Fourier model

SCA 99] SCARDI P., LEONI M., Fourier modelling of the anisotropic line broadening of x-ray diffraction profiles due to line and planar lattice defects , J. Appl. Cryst, vol. 32, p. 671-682, 1999. 

Fourier Model. High-resolulion image formation is considered most conveniently as occurring in two steps. In the first step, electrons propagate through the crystal and produce a two-dimensional periodic electron distribution at the exit face, which images the projected lattice potential or the projected electron density. In a second step, this exit face acts as a planar distribution of point sources of spherical electron waves, which interfere behind the foil and form the diffiacted beams that move in the lens sy.stem of the... 

A wave term represented by is present in equation (8.19). Here, y/ajr is a velocity-like quantity known as thermal wave speed C. Here, a and t are thermal properties of the Tnsteady state, the conduction equation reverts back to the Fourier model, even though the relaxation parameter, r 0. Therefore, temperature results for two models differ during the transient state only. 

Abstract Non-equilibrium thermodynamics offers a better and more precise way to describe transport of heat and mass over membranes than the simple Pick and Fourier models. In this work, we derive the equations for single component transport (of n-butane) across a zeolite membrane. We use non-equilibrium molecular dynamics (NEMD) data to show how transport phenomena are related. Significant differences are discussed between this method, and methods invoking Pick s and Fourier s laws. 

The probes are assumed to be of contact type but are otherwise quite arbitrary. To model the probe the traction beneath it is prescribed and the resulting boundary value problem is first solved exactly by way of a double Fourier transform. To get managable expressions a far field approximation is then performed using the stationary phase method. As to not be too restrictive the probe is if necessary divided into elements which are each treated separately. Keeping the elements small enough the far field restriction becomes very week so that it is in fact enough if the separation between the probe and defect is one or two wavelengths. As each element can be controlled separately it is possible to have phased arrays and also point or line focussed probes. 

We now proceed to some examples of this Fourier transfonn view of optical spectroscopy. Consider, for example, the UV absorption spectnun of CO2, shown in figure Al.6.11. The spectnuu is seen to have a long progression of vibrational features, each with fairly unifonu shape and width. Wliat is the physical interpretation of tliis vibrational progression and what is the origin of the width of the features The goal is to come up with a dynamical model that leads to a wavepacket autocorrelation fiinction whose Fourier transfonn... 

We will describe integral equation approximations for the two-particle correlation fiinctions. There is no single approximation that is equally good for all interatomic potentials in the 3D world, but the solutions for a few important models can be obtained analytically. These include the Percus-Yevick (PY) approximation [27, 28] for hard spheres and the mean spherical (MS) approximation for charged hard spheres, for hard spheres with point dipoles and for atoms interacting with a Yukawa potential. Numerical solutions for other approximations, such as the hypemetted chain (EfNC) approximation for charged systems, are readily obtained by fast Fourier transfonn methods... 

The experimental data and arguments by Trassatti [25] show that at the PZC, the water dipole contribution to the potential drop across the interface is relatively small, varying from about 0 V for An to about 0.2 V for In and Cd. For transition metals, values as high as 0.4 V are suggested. The basic idea of water clusters on the electrode surface dissociating as the electric field is increased has also been supported by in situ Fourier transfomr infrared (FTIR) studies [26], and this model also underlies more recent statistical mechanical studies [27]. 

A linear stability analysis of (A3.3.57) can provide some insight into the structure of solutions to model B. The linear approximation to (A3.3.57) can be easily solved by taking a spatial Fourier transfomi. The result for the Ml Fourier mode is... 

Fast Fourier Transformation is widely used in many fields of science, among them chemoractrics. The Fast Fourier Transformation (FFT) algorithm transforms the data from the "wavelength" domain into the "frequency" domain. The method is almost compulsorily used in spectral analysis, e, g., when near-infrared spectroscopy data arc employed as independent variables. Next, the spectral model is built between the responses and the Fourier coefficients of the transformation, which substitute the original Y-matrix. 

When applied to the rotation, vibration-rotation, or eleetronie-vibration-rotation eases within the "unhindered" rotation model treated earlier, the Fourier transform involves integrals of the form ... 

As mentioned, we also carried out IR studies (a fast vibrational spectroscopy) early in our work on carbocations. In our studies of the norbornyl cation we obtained Raman spectra as well, although at the time it was not possible to theoretically calculate the spectra. Comparison with model compounds (the 2-norbornyl system and nortri-cyclane, respectively) indicated the symmetrical, bridged nature of the ion. In recent years, Sunko and Schleyer were able, using the since-developed Fourier transform-infrared (FT-IR) method, to obtain the spectrum of the norbornyl cation and to compare it with the theoretically calculated one. Again, it was rewarding that their data were in excellent accord with our earlier work. 

AMPL. This framework is by Fourier and coworkers (Ref. 113) at Northwestern University. It is well suited for constructing complex models. 

Moews, P.C., Kretsinger, R.H. Refinement of the structure of carp muscle calcium-binding parvalbumin by model building and difference Fourier analysis. 

Fourier transform infrared (FTIR) spectroscopy is the most popular method for determining the imidization process in the solid state and identifying specific substituents on the macromolecular backbone (e.g., CN, SO3H, CO, SO2).131 A method for calculating the diermal imidization extent based on FTIR data has been reported by Pride.132 Raman spectroscopy was used on the model study of PMDA-ODA condensation, and the possible formation of an inline bond by reaction of an amino group with an imide carboxyle was evidenced.133... 

Keilson-Storer kernel 17-19 Fourier transform 18 Gaussian distribution 18 impact theory 102. /-diffusion model 199 non-adiabatic relaxation 19-23 parameter T 22, 48 Q-branch band shape 116-22 Keilson-Storer model definition of kernel 201 general kinetic equation 118 one-dimensional 15 weak collision limit 108 kinetic equations 128 appendix 273-4 Markovian simplification 96 Kubo, spectral narrowing 152... 

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