Orientation Averaging Research Articles

Dissociation of H+2 from a short, intense, infrared laser pulse: proton emission spectra and pulse calibration

The proton energy spectrum from photodissociation of the hydrogen molecular ion by short intense pulses of infrared light is calculated. The time-dependent Schrödinger equation is discretized and integrated. For few-cycle pulses one can resolve vibrational structure, arising from the experimental preparation of the molecular ion. We calculate the corresponding energy spectrum and analyse the dependence on the pulse time delay, pulse length and intensity of the laser for λ ∼ 790 nm. We conclude that the proton spectrum is a sensitive probe of both the vibrational populations and phases, and allows us to distinguish between adiabatic and nonadiabatic dissociation. Furthermore, the sensitivity of the proton spectrum from H+2 is a practical means of calibrating the pulse. Our results are compared with recent measurements of the proton spectrum for 65 fs pulses using a Ti:Sapphire laser (λ ∼ 790 nm) including molecular orientation and focal-volume averaging. Integrating over the laser focal volume, for the intensity I ∼ 3 × 1015 W cm−2, we find our results are in excellent agreement with these experiments.

Polyproline and the "spectroscopic ruler" revisited with single-molecule fluorescence.

To determine whether Forster resonance energy transfer (FRET) measurements can provide quantitative distance information in single-molecule fluorescence experiments on polypeptides, we measured FRET efficiency distributions for donor and acceptor dyes attached to the ends of freely diffusing polyproline molecules of various lengths. The observed mean FRET efficiencies agree with those determined from ensemble lifetime measurements but differ considerably from the values expected from Forster theory, with polyproline treated as a rigid rod. At donor-acceptor distances much less than the Forster radius R(0), the observed efficiencies are lower than predicted, whereas at distances comparable to and greater than R(0), they are much higher. Two possible contributions to the former are incomplete orientational averaging during the donor lifetime and, because of the large size of the dyes, breakdown of the point-dipole approximation assumed in Forster theory. End-to-end distance distributions and correlation times obtained from Langevin molecular dynamics simulations suggest that the differences for the longer polyproline peptides can be explained by chain bending, which considerably shortens the donor-acceptor distances.

Symmetric Euler orientation representations for orientational averaging

A new kind of orientation representation called symmetric Euler orientation representation (SEOR) is presented. It is based on a combination of the conventional Euler orientation representations (Euler angles) and Hamilton's quaternions. The properties of the SEORs concerning orientational averaging are explored and compared to those of averaging schemes that are based on conventional Euler orientation representations. To that aim, the reflectance of a hypothetical polycrystalline material with orthorhombic crystal symmetry was calculated. The calculation was carried out according to the average refractive index theory (ARIT [T.G. Mayerhöfer, Appl. Spectrosc. 56 (2002) 1194]). It is shown that the use of averaging schemes based on conventional Euler orientation representations leads to a dependence of the result from the specific Euler orientation representation that was utilized and from the initial position of the crystal. The latter problem can be overcome partly by the introduction of a weighing factor, but only for two-axes-type Euler orientation representations. In case of a numerical evaluation of the average, a residual difference remains also if a two-axes type Euler orientation representation is used despite of the utilization of a weighing factor. In contrast, this problem does not occur if a symmetric Euler orientation representation is used as a matter of principle, while the result of the averaging for both types of orientation representations converges with increasing number of orientations considered in the numerical evaluation. Additionally, the use of a weighing factor and/or non-equally spaced steps in the numerical evaluation of the average is not necessary. The symmetrical Euler orientation representations are therefore ideally suited for the use in orientational averaging procedures.

Elastoplastic Modeling of Metal Matrix Composites Containing Randomly Located and Oriented Spheroidal Particles

Micromechanics-based effective elastic and plastic formulations of metal matrix composites (MMCs) containing randomly located and randomly oriented particles are developed. The averaging process over all orientations upon three elastic governing equations for aligned particle-reinforced MMCs is performed to obtain the explicit formulation of effective elastic stiffness of MMCs with randomly oriented particles. The effects of volume fraction of particles and particle shape on the overall elastic constants are studied. Comparisons with the Hashin-Shtrikman bounds and Ponte Castaneda-Willis bounds show that the present effective elastic formulation does not violate the variational bounds. Good agreement with experimental elastic stiffness data is also illustrated. Furthermore, the orientational averaging procedure is employed to derive the overall elastoplastic yield function for the MMCs. Elastoplastic constitutive relations for the composites are constructed on the basis of the derived composite yield function. The stress-strain responses of MMCs under the axisymmetric loading are also investigated in detail. Finally, elastoplastic comparisons with the experimental data for SiCp/Al composites are performed to illustrate the capability of the proposed formulation.

Solid state nuclear magnetic resonance study of highly oriented poly(glycolic acid)

The structure of isotropic and highly drawn poly(glycolic acid) (PGA) fibres is investigated using solid-state 1H NMR spectroscopy. Relaxation times in the rotating frame ( T 1 ρ ) using a spin-lock method at 100 °C indicate that isotropic PGA can be adequately described by two phases, a rigid crystalline phase and an amorphous mobile phase with relaxation times of 16.6 and 0.9 ms, respectively. A crystallinity of 40% is obtained by this method and agrees well with previous calorimetric studies. In contrast, T 1 ρ measurements indicate that oriented PGA is 50% crystalline and can be adequately described by a three phase model consisting of: rigid crystalline material with long relaxation time, semi-rigid non-crystalline material with intermediate mobility and relaxation time and highly mobile non-crystalline material with the shortest relaxation time. It was found that the crystalline phase has a T 1 ρ = 53 and T 1 ρ = 39 ms when the fibre direction is at 0 and 90° relative to the external magnetic field, respectively. This difference in relaxation time is associated with higher spin interactions at 0°, reducing the effect on the mobility of the chains. The WAXS orientation averages P 2=0.99 and P 4=0.96 obtained from the azimuthal scans of the (020) and (002) reflections indicate a highly oriented crystalline structure. These results are used to contrast the structural information obtained from measurements and theoretical calculations of the rigid-lattice anisotropy of the second moment ( M 2), from which the orientation averages P 2=0.96 and P 4=0.94 were obtained on the basis of a published crystal structure. The discrepancies found are associated with small differences between the published crystal structure and that required to explain the spin interactions among adjacent molecules.

Structural analysis of clusters using configurational and orientational averaging in light scattering analysis

We estimate the morphological parameters of soot particles in flames from static light scattering measurements. For these measurements established methods based on X-ray crystal structure analysis produce erroneous results. As an alternative method we investigate orientational and configurational averaging methods in light scattering analysis and compare their results with that from TEM analysis of collected clusters.

Computational modeling of the response and damage behavior of fiber reinforced cellular concrete

A computational model is presented to predict the effective mechanical behavior and damage evolution in fiber reinforced cellular concrete (FRCC). The effective moduli of the FRCC are formulated based on the Eshelby's micromechanics and the orientational averaging process. The effects of random dispersion and orientation of inclusions are accommodated. Damage models are subsequently considered in accordance with the Weibull's probabilistic function to describe the varying probability of progressive fiber debonding and the continuum damage law proposed by Karihaloo and Fu [Eur. J. Mech. A 8 (1989) 62; ACI Mater. J. 87 (1990) 62] to model the nucleation of microvoids in the cement matrix. The constitutive model incorporating the damage mechanisms is then implemented into a finite element code to predict the performance of FRCC. The computational model is exercised to predict the performance of a typical fiber reinforced concrete to show the potential of the model. Finally, the prediction based on the computational model is compared with the experimental data to assess the capability of the model for predicting the performance of FRCC. The results show that the proposed computational model is a viable analytical and numerical tool for accurate evaluation of the performance of FRCC.

A Unified Treatment of Selection Rules and Symmetry Relations for Sum-Frequency and Second Harmonic Spectroscopies

A simplifying treatment is developed for describing the molecular origins of electric dipole allowed sum-frequency generation (SFG) and second harmonic generation (SHG). The full sum-over-states expressions for the nonlinear polarizability simplify tremendously at or near resonance to straightforward formulas easily connected to intuitive molecular properties. For resonance enhancement at the sum or second harmonic frequency, the molecular nonlinear polarizability tensor is shown to be the direct product of the transition moment and the two-photon absorption (TPA) polarizability tensor. To our knowledge, this is the first rigorous mathematical demonstration indicating such a simple relationship directly connecting second harmonic generation with TPA, providing a link between the two fields of inquiry. Under resonance enhancement with one of the incident frequencies, the SFG and SHG nonlinear polarizability tensors similarly are given by the products of the transition moments and the anti-Stokes Raman polarizability tensors (a reasonably well-known result for SFG). Under double-resonance conditions (i.e., resonant with one of the incident frequencies and the sum frequency), the two descriptions for the nonlinear polarizability become mathematically equivalent. Nonlinear optical character tables for both SHG and SFG under all resonance conditions have been compiled for chromophores of Cs, C2, C2v, and C3v symmetries. Explicit evaluation of the corresponding orientational averages for each allowed transition in each character table assuming a uniaxial macroscopic orientation distribution reveals numerous relationships connecting the microscopic symmetry with the macroscopic nonlinear response. The approaches developed in this work are sufficiently general to allow their use in interpreting electronic, vibrational, and vibronic spectroscopic measurements by SHG and SFG.

Edge preservation near triple junctions during orientation averaging of EBSP data

A feature-based methodology is given for evaluating the dependence of the modified Kuwahara filter on the orientation noise level. Inspection of simulated orientation maps of dislocation cell microstructures suggests a further modification that gives improved angular resolution whilst maintaining edge preservation near triple junctions.

Unified approach for the estimate of effective magnetostriction of composites and polycrystals with particulate and columnar microstructures

A unified approach for an estimate of the effective magnetostriction of composite systems is presented. Following an idea of Levin, originally devised for the estimate of effective thermal expansion of composite materials, we show that there exists an exact connection between the effective magnetostriction and influence functions for the composite system. The influence functions are defined as the average of local stress fields in each phase versus uniform stresses applied on the boundary. The exact formula states that the effective magnetostriction is spatial and orientational averages of local magnetostriction strain in which the influence functions serve as weighting factors. Various existing micromechanical approaches can be used to approximate the influence functions and thereby to provide an estimate for the effective magnetostriction. This result complements our previous findings that the effective magnetostriction can be directly linked to the effective elastic moduli for a few selected composite systems. Here an analogous link between magnetostriction and mechanical behavior can also be established, although more complex in algebra but much wider in the scope of applicability. For illustration, various micromechanical methods, such as dilute approximation, selfconsistent and Mori-Tanaka models, together with Voigt and Reuss assumptions, are employed to predict the effective saturation magnetostriction of composites. The estimates are compared with available experimental results and also with the reported theoretical estimates. The effects of particle shapes, orientations, and crystallite growth directions are illustrated for composite systems of technological interest.

Interpretation and modeling of IR-reflectance spectra of glasses considering medium range order

Conventional qualitative and quantitative IR-reflectance spectrum interpretation of glasses is discussed in light of new findings on the optics of polycrystalline materials and the close relation of the structure of crystallites and regions of medium range order in related glasses. According to these findings, a glass spectrum must not be compared in general with the spectrum of a related polycrystalline material, if the crystallite size exceeds the resolution limit of light. As a consequence of the similarities between the spectra of glassy and related polycrystalline compounds (optically small crystallites) and based on medium range order, the macroscopic optical properties of glasses result not only from disorder and fluctuations, but also from an orientational average of the optical properties of the medium range regions similar to the macroscopic optical properties of polycrystalline materials. Consequently, the assumption of cubic symmetry, which underlies all conventional types of dispersion analysis used for glasses, is inadequate. Based on ARIT (average refractive index theory), [Appl. Spectrosc. 56 (2002) 1194], which permits modeling the optical properties of polycrystalline materials with optically small crystallites, a method is proposed to generate artificial glass spectra from single crystal data. This method is particularly useful if polycrystalline bulk samples with optically small crystallites are not available, since it enables us to determine glass structure on a semi-quantitative level by comparing measured and artificial spectra. The value of the approach is demonstrated for fresnoite glass (Ba 2TiSi 2O 8), Sr-fresnoite glass (Sr 2TiSi 2O 8) and Ge-fresnoite glass (Ba 2TiGe 2O 8). An important consequence of the assumption of an orientational average of the microscopic optical properties of medium range order regions is the prediction of the occurrence of mixed TO–LO modes in glasses. This is confirmed by the resemblance between peak shapes of model oscillators with large oscillator strengths and small damping constants and the prominent high wave number feature of vitreous silica.

Optical properties of an eccentrically located pigment within an air bubble

We study the orientational average of the scattering efficiencies of scatterers composed by an eccentrically located spherical TiO 2 pigment within an air bubble embedded in resin. This study is carried out within a general T-matrix formalism and takes into account the contributions of the asymmetry parameter to the photon transport mean-free path. We perform a detailed comparison of the scattering efficiency of this system with the one corresponding to the same pigment particle embedded directly in resin. We showed that placing a TiO 2 pigment eccentrically into an air bubble embedded in resin could not increase the scattering efficiency of the pigment in comparison to a system consisting of the same fractional volume of pigment particles and the same fractional volume of air bubbles acting independently within the same resin.

A generalized self-consistent polycrystal model for the yield strength of nanocrystalline materials

Inspired by recent molecular dynamic simulations of nanocrystalline solids, a generalized self-consistent polycrystal model is proposed to study the transition of yield strength of polycrystalline metals as the grain size decreases from the traditional coarse grain to the nanometer scale. These atomic simulations revealed that a significant portion of atoms resides in the grain boundaries and the plastic flow of the grain-boundary region is responsible for the unique characteristics displayed by such materials. The proposed model takes each oriented grain and its immediate grain boundary to form a pair, which in turn is embedded in the infinite effective medium with a property representing the orientational average of all these pairs. We make use of the linear comparison composite to determine the nonlinear behavior of the nanocrystalline polycrystal through the concept of secant moduli. To this end an auxiliary problem of Christensen and Lo (J. Mech. Phys. Solids 27 (1979) 315) superimposed on the eigenstrain field of Luo and Weng (Mech. Mater. 6 (1987) 347) is first considered, and then the nonlinear elastoplastic polycrystal problem is addressed. The plastic flow of each grain is calculated from its crystallographic slips, but the plastic behavior of the grain-boundary phase is modeled as that of an amorphous material. The calculated yield stress for Cu is found to follow the classic Hall–Petch relation initially, but as the gain size decreases it begins to depart from it. The yield strength eventually attains a maximum at a critical grain size and then the Hall–Petch slope turns negative in the nano-range. It is also found that, when the Hall–Petch relation is observed, the plastic behavior of the polycrystal is governed by crystallographic slips in the grains, but when the slope is negative it is governed by the grain boundaries. During the transition both grains and grain boundaries contribute competitively.

Probing the Electronic Structure of Small Molecular Anions by Photoelectron Imaging

We outline the methodology of negative-ion photoelectron imaging and general aspects of interpretation of the results using the CS2- and S2- anions as model systems. The CS2- images are recorded using 800, 530, 400, and 267 nm photons. The observed transitions result in the formation of CS2 in the X 1Σg+, a 3B2, b 3A2, and A 1A2 states. The S2- measurements are carried out at the same wavelengths with the exception of 800 nm. The resulting images reveal the detachment transitions assigned to the X 3Σg-, a 1Δg, b 1Σg+, c 1Σu-, and A‘ 3Δu states of the neutral. The choice of detachment wavelengths serves as a “zoom” selectively focusing on chosen transitions, in some cases allowing the observation of their vibrational structure. The photoelectron spectra and angular distributions obtained from the images are used to discuss the electronic structure and detachment dynamics. In particular, two approaches to interpreting the angular distributions are discussed. One method employs the Cooper−Zare central-potential model adapted to the molecular case. It considers an expansion of the parent orbital in the basis of single-center atomic-orbital functions, for which the partial waves comprising the ejected electron are determined. The application of this model to molecular anions is straightforward, if the parent molecular orbital resembles an atomic orbital, which is the case for S2-, but not CS2-. In the latter case, a different qualitative approach is proposed, which (i) relies upon the electric-dipole approximation and group theory for the determination of the detached electron wave function symmetry, (ii) restricts the analysis to symmetry (electric dipole) allowed s and p partial waves, and (iii) qualitatively treats the orientation averaging by considering only a few “principal” molecular orientations. The results provide a foundation for the qualitative interpretation of anion photoelectron images.

Orientation-dependent integral equation theory for a two-dimensional model of water

We develop an integral equation theory that applies to strongly associating orientation-dependent liquids, such as water. In an earlier treatment, we developed a Wertheim integral equation theory (IET) that we tested against NPT Monte Carlo simulations of the two-dimensional Mercedes Benz model of water. The main approximation in the earlier calculation was an orientational averaging in the multidensity Ornstein–Zernike equation. Here we improve the theory by explicit introduction of an orientation dependence in the IET, based upon expanding the two-particle angular correlation function in orthogonal basis functions. We find that the new orientation-dependent IET (ODIET) yields a considerable improvement of the predicted structure of water, when compared to the Monte Carlo simulations. In particular, ODIET predicts more long-range order than the original IET, with hexagonal symmetry, as expected for the hydrogen bonded ice in this model. The new theoretical approximation still errs in some subtle properties; for example, it does not predict liquid water’s density maximum with temperature or the negative thermal expansion coefficient.

Modelling IR-spectra of single-phase polycrystalline materials with random orientation – Supplementations and refinements for optically uniaxial crystallites

Summary The IR-reflectance spectra of a bulk sample of randomly oriented polycrystalline fresnoite (Ba2TiSi2O8) are presented. The sample was prepared by Spark Plasma Sintering from a powder consisting of optically large crystallites. The spectra are of notable higher quality than those of conventionally prepared samples, especially those of the cross-polarization terms. The spectra were modelled by a 4 ×4-matrix-formalism [T. G. Mayerhofer, J. Opt. A: Pure Appl. Opt. 4, 540 (2002)], which was refined for optically uniaxial crystals, resulting in a correct prediction of the relative peak-intensities of the cross-polarization terms. The modelling was carried out with two different orientational averaging schemes. The comparison between modelled and measured spectra revealed that the use of any weighing factor is unjustified in case of orientational averaging of the IR-optical properties of solids.

Orientation-averaged radiative properties of an arbitrary configuration of scatterers

We discuss both analytical and numerical methods in calculating radiative scattering parameters of a randomly oriented ensemble of particles. There exist two approaches to rigorously solving multiparticle scattering. One is the generalized multiparticle Mie-solution (GMM) and the other is cluster T-matrix approach (CTM). We show that, the same as in the case of fixed-orientation scattering, the random-orientation formulations of GMM and CTM have a remarkable contrast in capability of handling the overall dimension of an ensemble of particles in practical applications. We also show that the use of a standard scheme of numerical quadrature could be more efficient than straightforwardly taking an unweighted arithmetic average in discrete orientational averaging.

Orientation Averaging for Stiffness and Thermal Expansion of Short Fiber Composites

It is fairly common in practice that during injection moulding, the mold filling process results in non‐uniform fiber orientation distributions in the final injection molded short fiber reinforced composite part. It would be very attractive to employ an orientation averaging scheme to form predictions for the design of short fiber reinforced composite parts. Here, the authors use a finite‐element‐based numerical procedure developed by themselves and, for the first time, directly predict the stiffness and thermal expansion of several hundred multi‐fiber computer models with a variety of different predefined fiber orientation states.

Implementation of the T-matrix method on a massively parallel machine: a comparison of hexagonal ice cylinder single-scattering properties using the T-matrix and improved geometric optics methods

This paper describes a series of improvements to an already existing T-matrix implementation for calculating the single-scattering properties of non-axisymmetric particles at large size parameters. The improvements aim to use the computational resources in terms of memory and time as efficiently as possible by taking into account various simplifications of the T-matrix formulation. In this paper, the randomly oriented hexagonal ice crystal is considered and the orientational averaging is done analytically. Within the framework of the T-matrix approach, symmetries of the hexagonal cylinders allow the formulation of eight independent subproblems, for each of which the memory demand is reduced by a factor of 144 compared to the general formulation. The memory demand strongly increases as a function of size parameter. For non-axisymmetric particles, memory demand rises approximately as the fourth power of the size parameter. In order to have more memory available, the T-matrix code was parallelized and implemented on a multi-processor CRAY T3E supercomputer. Extended precision is software emulated. With these new developments the single-scattering properties of randomly oriented hexagonal columns can be calculated for size parameters up to about 40. This size parameter reaches into the region of improved geometric optics (IGO), which is applicable down to size parameters of around 20. Comparisons of IGO and T-matrix results for the single-scattering albedo and the extinction efficiency show that at a size parameter of around 30, the single-scattering albedos are generally within about 2%. IGO underpredicts the extinction efficiency by 4% at λ=3.775 μm and by 11% at the more absorbing wavelength of λ=4.9 μm . This underprediction may be due to the version of IGO used since this does not take into account the inhomogeneity of the refracted wave in an absorbing medium.

A transfer matrix approach to local field calculations in multiple-scattering problems

We present a new recursive transfer matrix method for calculating local electromagnetic fields in systems of spheres subject to strong dependent scattering. Local field information is often lost or discarded in recursive transfer matrix approaches. In order to preserve the local field information, and to avoid problems associated with the dimensional cut-off of the translation matrices, we calculate the scatterer-centred transfer matrices. Our technique permits systematic studies of local field eiects for all possible incident field directions, and configurations (including orientation averages). Illustrative calculations are presented.