Moment Scheme Research Articles

Measurement of bipolar moments for photofragment angular correlations in ion imaging experiments

A general numerical method is given to extract angular correlations from photodissociation experiments with ion imaging detection. The angular correlations among the transition dipole moment of the parent molecule, μ, the photoproduct recoil velocity, v, and its angular momentum, j, are parametrized analytically using the semiclassical bipolar moment scheme due to Dixon. The method is a forward-convolution scheme which allows quantitative extraction of all measurable bipolar moments and can be applied in experiments with both linearly and circularly polarized probe light. It avoids the cylindrical symmetry limitations of the inverse Abel transform method, traditionally used for extracting photoproduct recoil anisotropy and speed distribution from imaging data. The method presented here also takes into account the possibility of multiple photodissociation channels. The features of the method are illustrated in a two-color 1+1′ REMPI-ion imaging study of the NO photoproduct trajectories resulting from the 650 nm photodissociation of 2-chloro-2-nitrosopropane (CNP). A comparison between experimental and synthetic images is presented for selected experimental geometries. The experimental images for CNP and the results from their fit confirm earlier TOF studies showing that the recoil speed distribution is bimodal with the low and high speed components having average values of approximately 500 and 910 m/s. These components have been previously assigned to dissociation from the S0 and T1 electronic states of the parent molecule, respectively. The experimental results from the current study also confirm that for the high-speed component the product NO velocity vector, v, is preferentially perpendicular to its angular momentum, j [β00(22)=−0.21], and that there is no significant correlation between v and the transition dipole moment μ of the CNP molecule [β02(20)=−0.02].

Semileptonic decays of the B c meson

The semileptonic decays of the B c meson into heavy quarkonia J/ψ(ηc) and a pair of leptons are investigated on the basis of three-point sum rules of QCD and nonrelativistic QCD (NRQCD). Calculations of analytic expressions for the spectral densities of QCD and NRQCD correlation functions with allowance for Coulomb-like αs/v terms are presented. At recoil momenta close to zero, generalized relations due to the spin symmetry of NRQCD are derived for the B c → J/ψ(ηc)/νl form factors, with l denoting one of the leptons (e, μ, or τ). This allows one to express all NRQCD form factors in terms of a single universal quantity, an analog of the Isgur-Wise function at the maximal lepton-pair invariant mass. The gluon-condensate corrections to three-point functions are calculated both in full QCD in the Borel transform scheme and in NRQCD in the moment scheme. This enlarges the parametric-stability region of the sum-rule method, thereby rendering the results of the approach more reliable. Numerical estimates of the widths for the transitions B c → J/ψ(ηc)lνl are presented.

Semileptonic Bc-meson decays in sum rules of QCD and NRQCD

The semileptonic B c -meson decays into the heavy quarkonia J/ ψ( η c ) and a pair of leptons are investigated in the framework of three-point sum rules of QCD and NRQCD. Calculations of analytical expressions for the spectral densities of QCD and NRQCD correlators with account for the Coulomb-like α s / v terms are presented. The generalized relations due to the spin symmetry of NRQCD for the form factors of B c → J/ ψ( η c ) lν l transitions with l denoting one of the leptons e, μ or τ, are derived at the recoil momentum close to zero. This allows one to express all NRQCD form factors through a single universal quantity, an analogue of Isgur–Wise function at the maximal invariant mass of lepton pair. The gluon condensate corrections to three-point functions are calculated both in full QCD in the Borel transform scheme and in NRQCD in the moment scheme. This enlarges the parameteric stability region of sum rule method that makes the results of the approach to be more reliable. Numerical estimates of widths for the transitions of B c → J/ ψ( η c ) lν l are presented.

Moment scheme of the finite-element method in problems of the strength and stability of flexible shells subjected to the action of forces and thermal factors

We propose a method for investigation of the stability of shells under the action of forces and thermal factors that is based on relations of the geometrically nonlinear theory of elasticity and the moment scheme of finite elements. For the case of uniformly heated restrained cylinders, we demonstrate the efficiency of the method and the validity of the results obtained. We analyze the distinctive features of deformation of the investigated class of shells.

Evaluation of the stress intensity factor for cracks in elastomers

We analyze the applicability of the method of the J-integral to the solution of problems of fracture mechanics for structures made of low-compressible elastomers. The components of the J-integral are computed by the method of equivalent three-dimensional integration. To take into account weak compressibility, we use the finite-element moment scheme.

Numerical simulation of temperature fields in elastomeric structures with cracks on the basis of singular finite elements

We describe a method used for the investigation of temperature fields of dissipative heating in elastomeric structures with cracks. Determining relations are constructed on the basis of a moment scheme of finite elements with application of singular quadratic elements. The\({1 \mathord{\left/ {\vphantom {1 {\sqrt r }}} \right. \kern-\nulldelimiterspace} {\sqrt r }}\)-type singularities of the stress and strain fields at the crack tip are simulated by shifting the intermediate nodes by a quarter of the length of the side toward the crack tip in the basic coordinate system, while the functions of shape remain unchanged in the process of construction of the stiffness matrix. As an example, we study the problem of determination of temperature fields in continuous cylindrical, trough-shaped, and rectangular shock absorbers of complex geometry with cracks. The proposed method is realized in the form of a software complex for personal computers called KODETOM.

Stability of Nonlinear Two-Frequency Oscillation of Cylindrical Shells

The moment scheme of the finite element method and the method of generalized coordinates are used to construct a multi-degree-of-freedom nonlinear model of a cylindrical shell subjected to two-frequency excitations. This model consists of a system of nonlinear differential equations. The incremental method is then used to find the solution of the equation in the frequency domain, while the Poincare map, spectral analysis, and Floquet's theory are applied to the stability of the solution at every step of the incremental method. Solutions and discussions are presented to substantiate the suggested algorithm. It is shown that similar results are obtained by using the Poincare map with numerical integration and Floquet's theory with Fourier's expansion. However, Floquet's theory is a lot less time-consuming and it pinpoints more accurately the moment of loss of stability.

Computer simulation of the processes of mechanical shaping of three-dimensional shells under the conditions of creep. Part 2

On the basis of the moment scheme, we obtain the resolving relations of the method of finite elements in the actual configuration used for the computer simulation of the processes of shaping of spatial shells by using a single finite element across the thickness. The reliability of numerical results is checked by the solution of a test example of shaping of a long narrow rectangular membrane. As an illustration of the possibilities of the developed approach to the analysis of the processes of shaping of thin-walled products with complicated shape, we present the results of numerical simulation of strains in a blank made of D16 alloy in a discrete press tood.

The Discretized Mie-Formalism for Electromagnetic Scattering - Summary

In this contribution, the Discretized Mie-Formalism (DMF) for plane wave scattering by irregularly shaped objects is presented in both cylindrical and spherical co-ordinates. Besides the direct method we discuss an iterative procedure which results in less computational effort and a higher numerical stability. This makes the DMF applicable to aspect ratios and size parameters which have only been treated by use of the Extended Boundary Condition Method so far. The iterative DMF is a Method of Moment scheme applied to the continuity condition of the tangential electromagnetic field components at the scatterer surface. Based on the differential equation formulation, the DMF can be considered to be the numerical generalization of the Mie theory to particles with non-spherical boundaries. By use of a special Finite Difference technique, the so-called Method of Lines, the Helmholtz equations for the Debye potentials, related to the scattering process, can be transformed into an uncoupled system of ordinary differential equations depending only on the radial coordinate. This system can be solved analytically by taking the radiation condition into account. In this way, the final calculation can be restricted to the surface of the scatterer as known from surface integral approaches. Additionally, a decoupling of the orientation of the scatterer with respect to the incident field from its geometrical and physical configuration is achieved. The theoretical description is completed by an application to scatterers with different shapes, and in different size parameter regions.

Analysis of Nuclear Relaxation Experiments in HighTcOxides Based on Mila-Rice Hamiltonian

Nuclear relaxation experiments of 63 Cu, 17 O and 89 Y NQR/NMR in high T c oxides have been analyzed within the frame of a phenomenological model proposed by Mila and Rice, supposing that only one spin freedom exists. It was shown quantitatively that 17 (1/ T 1 ), 89 (1/ T 1 ) and the approximate T -linear behavior of 63 (1/ T 1 ) at higher temperatures can be associated with the same d-spin fluctuations around q ∼0. The anisotropy data of 63 (1/ T 1 ) were also explained consistently, from which the values of hyperfine coupling constants were determined. The contribution of antiferromagnetic d-spin fluctuations to 63 (1/ T 1 ) was extracted from raw data without any adjustable parameters . The saturated value of it above ∼150 K is by an order of magnitude lower than that estimated in a localized moment scheme. A possible interplay between magnetism and superconductivity is also discussed.

Projection operator analysis of macrotransport processes

Projection operator methods are used to derive a systematic perturbation expansion scheme that provides formal justification for the otherwise ad hoc arguments underlying generalized Taylor dispersion theory. The latter theory is a coarse-graining procedure for eliminating the internal variables from multidimensional phase-space convective-diffusion-type transport equations in the limit of long times. In addition to rationalizing the otherwise ad hoc Lagrangian moment scheme, projection operator methods are further used to: (i) investigate nonasymptotic effects arising from ‘‘memory’’ of the initial conditions; (ii) establish the existence of higher-order contributions beyond second-order terms in the physical-space spatial gradients. Explicit criteria are developed for when these third- and higher-order non-Gaussian terms may be neglected in the coarse-grained, Taylor dispersion or macrotransport equation.

Construction of resolvent finite-element equations for problems of the theory of elasticity. Report no. 2

1. The proposed scheme for obtaining resolvent FEM equations, based on approximation of the divergence and curl of the displacement vector by element, is consistent with the classical FEM scheme. 2. Having the advantages of the FEM moment scheme [6], the divergence-curl scheme permits simplification of the stiffness matrix by reducing volume integration to surface integration. It also presents the problem of optimizing the stiffness matrix of elements with a certain number of unknowns from premises of the Melosh hypothesis. 3. The properties of monotonic energy convergence of the divergence-curl scheme makes it possible to use it in calculations of various problems (beams, slabs, shells, etc.) with one type of element, which is very important in developing branched program packs of universal application.

Taylor dispersion in systems of sedimenting nonspherical Brownian particles: III. Time-periodic forces

An Aris-type moment scheme is employed to calculate the time-average mean velocity vector and Taylor-Aris dispersion tensor for the Stokes sedimentation of small homogeneous ellipsoidal (and other orthotropic) particles which simultaneously undergo translational and rotational Brownian motions while settling in an otherwise quiescent viscous fluid under the influence of a time-periodic external force. This generalizes to time-periodic forces prior dispersion results previously restricted to time-independent forces. Each harmonic component of the force independently gives rise to its own Taylor dispersion contribution, in which appears a dimensionless coefficient quantifying the effects of frequency and particle geometry—namely size and shape. This parameter vanishes identically for spherical and other hydrodynamically isotropic particles (e.g., cubes, tetrahedra, octahedra, etc.), whose translational hydrodynamic resistances are orientation independent. Advantages attaching to an experimental “wet” particle-shape characterization device based upon the principles developed herein are pointed out. In this context a novel experimental device is proposed based upon preventing any net (i.e., time-average) gravitational sedimentation of the nonneutrally buoyant particles by continuous rotation of the suspension about a horizontal axis.

Taylor dispersion in systems of sedimenting nonspherical Brownian particles: II. Homogeneous ellipsoidal particles

An Aris-type moment scheme is applied to calculate the Taylor-Aris dispersion tensor for the sedimentation of small homogeneous ellipsoidal (and other orthotropic) particles settling under the influence of gravity in a quiescent viscous fluid and undergoing rotational and translational Brownian motions. This generalizes to triaxial particles a prior dispersion result for centrally symmetric bodies of revolution, such as spheroids. An independent Langevin-type dispersivity calculation is shown to yield results identical to those obtained by the moment scheme. The components D ̄ |∗ and D ̄ ⊥∗ of the transversely isotropic dispersion dyadic, parallel and perpendicular, respectively, to the direction of the gravitational field, are shown to be given by the Taylor—Aris-type formulas D ∥ ∗ = D+ γ U ∗2 15 d , D ⊥ ∗ = D+ γ U ∗2 20d . Here, U ̄ ∗ is the mean settling velocity of the particle, and D ̄ = ( 1 3 )(D 1 + D 2 + D 3) and d ̄ = ( 1 3 ) × (d 1 + d 2 + d 3) are respectively the mean translational and rotational diffusivities of the Brownian particle, with (D 1,D 2,D 3) and (d 1,d 2,d 3) respectively the appropriate diffusivity components along the principal axes of the particle. The dimensionless coefficient γ, which is of order unity, is given by the formula γ= d[( D 1 D−1 ) 2d 1+( D 2 D−1 ) 2d 2+( D 3 D−1 ) 2d 3] d 1d 2+d 2d 3+d 3d 1 , This anisometric parameter vanishes identically for spherical particles and other hydrodynamically isotropic particles (e.g., cubes, tetrahedra, octahedra, etc.) whose translational and rotational hydrodynamic resistances are independent of the orientation of the particle relative to the directions of its linear and angular velocity vectors. Upon utilizing the translational and rotational Stokes-Einstein equations, explicit numerical values of D ̄ |∗ and D ̄ ⊥∗ are furnished for ellipsoids of revolution of various aspect ratios and sizes when settling in water. Physical restrictions pertaining to sedimentation-vessel apparatus size and the requirement of reasonable sedimentation times greatly restrict the range of particle sizes whose anisometric properties may be experimentally investigated by this new particle-shape characterization technique.

Polar bonds and polarizability in conformational energy calculations: Application of a polarization model to alkyl halides

Conformational energy models traditionally attempt to simulate molecular properties by means of a total molecular energy assembled from empirically determined transferable energy functions that include bond stretching, bending, and torsion as well as nonbonded interactions. In extending the model to polar molecules through the addition of bond-centered dipoles or atom-centered charges, questions arise concerning inductive effects on assumed bond moments and the effect of intervening atoms on the electrostatic interaction between polar groups. We investigate here the use of a polarization–mutual induction model in representing these effects. Each bond is assumed to carry an intrinsic dipole moment and a polarization center. The induced moment at each polarization center is computed from the electric field arising from all sources (the other permanent and induced moments in the molecule and an external field). The model is parameterized for calculations on haloalkanes by determining intrinsic C–X bond monents, C–X, C–H, and C–C bond polarizabilities as well as stretching and bending constants involving halo atoms from experimental data. Other C, H parameters are taken from a previous hydrocarbon force field. It is found that molecular polarizabilities and dipole moments are reproduced very well including moments in molecules where inductive effects render a fixed bond moment scheme inaccurate. Conformational energies are generally satisfactorily accounted for. Electrostatic contributions to the latter are found to be important in determining conformer stabilities.

Theory of interface energies

We have developed a practical scheme for the calculation of interface energies. It combines the theory of generalized Wannier functions, the generalized recursion method for calculating local densities of states and electronic density, and the local-density-functional theory. As a first application of this method we are calculating the stacking-fault energy of nickel using tight-binding-type Wannier functions for the $d$ electrons and ignoring the effect of the $s$ electrons. The $d$-band degeneracy is fully taken into account. The method also allows one to handle charge-transfer effects: a stacking-fault perturbation potential is included and calculated self-consistently. Comparison is made between the moment scheme, the non-self-consistent scheme, and the self-consistent scheme. Earlier moment calculations, up to the fourth moment, lead to stacking-fault energies which are a factor of 3 smaller than our more complete calculations. We also find that the self-consistent refinement affects the stacking-fault energy very little. Our best theoretical result is still much smaller, by a factor of about 3-7, than the (rather uncertain) experimental energy.

Moment scheme of the finite-element method in geometrically nonlinear problems regarding the strength and stability of shells

1. The finite-element method may be successfully and effectively used in solving problems regarding the strength and stability of sloping and nonsloping shells and plates. The rejection of the simplifying hypotheses made in the theory of shells and also the allowance made for the rigid displacement of the body enables the results of the theory of thin plates and shells to be refined, the limits of its applicability to be established, and a faster convergence of the results to be achieved as compared with other versions of the finite-element method. 2. The algorithm employed for the solution of geometrically nonlinear problems offers the possibility of studying the hypercritical behavior of sloping and nonsloping shells in the best possible way. 3. The version of the MSFE here described may also be used in solving physically nonlinear problems and problems regarding the dynamics of plates and shells.