Effect of surface modes on the six-dimensional molecule-surface scattering dynamics of H2-Cu(100) and D2-Cu(111) systems

We include the phonon modes originating from the three layers of Cu(100)/Cu(111) surface atoms on the dynamics of molecular [H(2)(v,j)/D(2)(v,j)] degrees of freedom (DOFs) through a mean field approach, where the surface temperature is incorporated into the effective Hamiltonian (potential) either by considering Boltzmann probability (BP) or by including the Bose-Einstein probability (BEP) factor for the initial state distribution of the surface modes. The formulation of effective potential has been carried out by invoking the expression of transition probabilities for phonon modes known from the "stochastic" treatment of linearly forced harmonic oscillator (LFHO). We perform four-dimensional (4D?2D) as well as six-dimensional (6D) quantum dynamics on a parametrically time and temperature-dependent effective Hamiltonian to calculate elastic/inelastic scattering cross-section of the scattered molecule for the H(2)(v,j)-Cu(100) system, and dissociative chemisorption-physisorption for both H(2)(v,j)-Cu(100) and D(2)(v,j)-Cu(111) systems. Calculated sticking probabilities by either 4D?2D or 6D quantum dynamics on an effective potential constructed by using BP factor for the initial state distribution of the phonon modes could not show any surface temperature dependence. In the BEP case, (a) both 4D?2D and 6D quantum dynamics demonstrate that the phonon modes of the Cu(100) surface affect the state-to-state transition probabilities of the scattered H(2) molecule substantially, and (b) the sticking probabilities due to the collision of H(2) on Cu(100) and D(2) on Cu(111) surfaces show noticeable and substantial change, respectively, as function of surface temperature only when the quantum dynamics of all six molecular DOFs are treated in a fully correlated manner (6D).     

The effect of phonon modes on the H2(v, j)/D2(v, j)-Cu(1nn) scattering processes

We include the effect of the phonon modes originating from the three layers of Cu(1nn) surface atoms on the dynamics of incoming molecular [H(2)(v, j)/D(2)(v, j)] degrees of freedom (DOFs) through a mean-field approach, where the surface temperature is incorporated into the effective potential by considering Bose-Einstein probability (BEP) factor for the initial state distribution of the surface modes calculated within harmonic approximation. Such time and temperature dependent effective Hamiltonian is further subdivided assuming a weak coupling between the two sets of molecular DOFs, namely, (x, y, z, Z) and (X, Y), respectively, in particular, to reduce the computational cost and the corresponding coupled quantum dynamical equations of motion have been formulated in terms of Time Dependent Discrete Variable Representation (TDDVR) approach. We demonstrate the workability of TDDVR method to investigate the scattering of H(2)(v, j) on Cu(1nn) surface by calculating the reaction probabilities and scattering cross-sections. Calculated results show that the phonon modes affect (a) the state-to-state transition probabilities of the scattered H(2) molecule substantially but chemisorption and physisorption processes negligibly and (b) the reaction probability of the incoming D(2) molecule noticeably.     

Third-order nonlinear optical studies of newly synthesized polyoxadiazoles containing 3,4-dialkoxythiophenes

The third-order nonlinear optical properties of newly synthesized soluble polyoxadiazoles containing 3,4-dialkoxythiophenes were investigated by using Z-scan and degenerate four-wave mixing (DFWM) techniques. The measurements were performed at 532 nm with 7 ns pulses from a Nd:YAG laser. We found a good agreement between the values of χ⁽³⁾ determined from both the experiments. Z-scan results indicate a negative nonlinear refractive index, n₂, whose magnitude is of the order of 10⁻¹⁰ esu for all the copolymers. The variation of nonlinear response among the copolymers clearly indicates the dependence of χ⁽³⁾ on donor and acceptor type of units present in these copolymers. The copolymers exhibit strong reverse saturable absorption and good optical limiting properties at 532 nm.     

Absence of Cooper-type bound states in three- and few-electron systems

It is shown that the appearance of a fixed-point singularity in the kernel of the two-electron Cooper problem is responsible for the formation of the Cooper pair for an arbitrarily weak attractive interaction between two electrons. This singularity is absent in the problem of three and few superconducting electrons at zero temperature on the full Fermi sea. Consequently, such three- and few-electron systems on the full Fermi sea do not form Cooper-type bound states for an arbitrarily weak attractive pair interaction. Received: 9 February 1998 / Revised and Accepted: 13 May 1998     

Tracking noisy limit cycle oscillation with nonlinear filters

For engineering systems, the dynamic state estimates provide valuable information for the detection and prediction of failure due to noise and vibration. From this perspective, nonlinear filtering techniques are applied to the problem of state estimation of dynamical systems undergoing noisy limit cycle oscillation. Specifically, the extended Kalman filter, ensemble Kalman filter and particle filter are used to track the limit cycle oscillations of a Duffing oscillator using noisy observational data. The noisy limit cycle oscillations feature highly non-Gaussian trends. The efficiency and limitations of the extended Kalman filter, ensemble Kalman filter and particle filter in tracking limit cycle oscillations are examined with respect to the model and measurement noise and sparsity of measurement data. For the limit cycle oscillations considered here, it is demonstrated that the ensemble Kalman filter and particle filter outperform the extended Kalman filter in the presence of sparse observational data or strong measurement noise. For moderate measurement noise and frequent measurement data, the ensemble Kalman filter and particle filter perform equally well in comparison to the extended Kalman filter.     

The 2D Boussinesq equations with vertical viscosity and vertical diffusivity

This paper aims at the global regularity of classical solutions to the 2D Boussinesq equations with vertical dissipation and vertical thermal diffusion. We prove that the Lr-norm of the vertical velocity v for any 1<r<∞ is globally bounded and that the L^∞-norm of v controls any possible breakdown of classical solutions. In addition, we show that an extra thermal diffusion given by the fractional Laplace δ(−Δ) for δ>0 would guarantee the global regularity of classical solutions.     

Evaluation of Morphology and Deformation Micromechanisms in Multilayered PP/PS Films: An Electron Microscopy Study

Nanostructured polymers and ultra-thin polymer layers are used more and more in technical applications like nanotechnology and microelectronics. Therefore, it is really important to understand the size-scale dependent properties as bulk polymers become thinner and more two-dimensional. Here the morphology as well as the macroscopic and the microscopic deformation behaviour of multilayered films of polypropylene (PP) and polystyrene (PS) have been investigated. For investigation different microscopic techniques and tensile testing were used. The films were prepared by multilayer coextrusion, whereas the composition of PP and PS and the film thickness – and therefore the thickness of each layer – varied. The thinnest calculated thickness of a single layer was about 5 nm. It is shown that the PP/PS films consist usually of homogeneous layers with only few defects. As the composition of PP/PS deviates strongly from 50/50 or the films get thinner the number of defects increases and the layered system turns to irregular lamellar system. In macroscopic tensile tests the small PS layers affect the elongation at break enormously: Most of the samples are brittle. For the films with a composition of PP/PS 90/10 and the film PP/PS 70/30 with a film thickness of 25 µm an elongation at break of 66% and higher could be reached. Transmission electron microscopy on these samples shows that the layers are characterized by plastic yielding in local deformation zones.     

Zero-sum problems with subgroup weights

In this note, we generalize some theorems on zero-sums with weights from [1], [4] and [5] in two directions. In particular, we consider ℤ _p ^d for a general d and subgroups of Z* _p as weights.