Kinematic approach

Structure refinement based on kinematical scattering was already applied by the Russian scientist 60 years ago. Weirich et al. (1996) first solved the structure of an unknown TinSe4by HREM combined with crystallographic image processing. Then they used intensities extracted from selected area electron diffraction patterns of a very thin crystal and refined the structure to a precision of 0.02 A for all the atoms. Wagner and Terasaki et al. (1999) determined the 3D structure of a new zeolite from selected area electron diffraction, based on kinematical approach. 

Rigorous modeling is the desired ideal. This means the ability to develop a complete process model with no direct measurement of process values. This kind of modeling generally relies on balance equations, kinematic approaches (e. g., for transference of energy by thermal conduction) and material data. 

The kinematical approach is simple, and adequately and accurately describes the diffraction of x-rays from mosaic crystals. This is especially true for polycrystalline materials where the size of crystallites is relatively small. Hence, the kinematical theory of diffraction is used in this chapter and throughout this book. 

Although this concrete explanation of the amplitude contrast is appropriate for a basic understanding, it does not account for all image features. It is based on the kinematical approach to explain electron diffraction. In the kinematical approach, an electron can be scattered once and once it is scattered it will not change its momentum. For very thin samples of light elements, this approximation is sometimes justifiable. 

When the SLD profile of an interface is known, a matrix method or a recursion method can easily be used to calculate reflectivity curves. However, for pedagogical purposes the relationship between the reflectivity and the structure of the interface is better revealed by the analytical expressions derived with the help of the kinematic approximation. The kinematic approximation has been shown to describe the reflection of neutrons from stratified media very well when the reflectivity is significantly less than unity. When a film of SLD pi and thickness t, is sandwiched between two phases of identical SLD p, the expression for the reflectivity derived from a kinematic approach is [36]... 

The above discussion reveals how NR can be used to provide direct information concerning both the thickness and the composition via the SLD of the various layers in the interface. This constitutes an advantage over ellipsometry or surface plasmon resonance methods where the value of the refractive index (composition) of the film is usually assumed in order to determine its thickness. However, to use the kinematic approach to directly determine these quantities, the reflectivity has to be measured in a sufficiently large range of so that at least one-half of the longest period interference fringe, corresponding to the thirmest layer in the interface, is observed on the reflectivity curve. The spatial resolution for a film of thickness ris defined as t= 7r/Qz,max, where is the maximum momentum... 

These two approximations are what characterizes the kinematic approach to diffraction and make it possible to describe the effect in a relatively simple way. The dynamic theory [AUT 05] takes both of these effects into account. The implementation of this dynamic theory is necessary when studying diffraction by very high qtrality single crystals, or also in the field of homoepitxial thin film characterization. These two subjects are beyond the scope of this book and therefore from here on, we will apply the kinematic theory of X-ray diffraction. 

There are two general theories that describe observed intensities in x-ray diffraction these are the kinematical and dynamical theories. The kinematical approach, which is the better known and most commonly employed, treats the scattering from each volume element independently of each other except for the incoherent power losses associated with the beam reaching and leaving a particular volume element. 

The alternate theory, called the dynamical theory of x-ray diffraction, takes into account all wave interactions within a crystal and must be used when considering diffraction from perfect or nearly perfect crystals. The dynamical approach represents a general theory to account for observed intensities in x-ray diffraction, and the kinematical approach can be considered one of its limiting cases. For small diffracting volumes and weak reflections, both theories predict the same results. However, when considering perfect or nearly perfect crystals and strong reflections, the kinematical approach gives an incorrect account of the diffracted intensities, and the dynamical approach must be employed. 

The kinematic treatment is generally used to interpret surface diffraction data. However, one must remember that the total external reflection effect is dynamial in nature so that the kinematic treatment is strictly not applicable. However, as has been pointed out by Vineyard, a simple distorted wave approximation ean be used, and this can be quite adequately treated in a kinematical approach. This last point greatly simplifies data interpretation. 

In this chapter, we use the results of numerical infiltration experiments in dual porosity media performed with a three-dimensional lattice-gas model to characterize preferential flow as response to rainfall intensity. From the temporal and spatial evolution of the water content during infiltration and drainage, we evaluate the adequacy of a kinematic wave approximation to describe the flow. We also discuss the conceptual basis of the asymptotic kinematic approach to Richards equation in comparison with the macropore kinematic equation. 

As discussed by Germann and Di Pietro (1996) the kinematic wave model applies in two modes. The 9-mode presented above and the J-mode that is obtained by writing Eq. [26] in terms of J using relation 23. To test the validity of the kinematic approach, the J-mode allows for input-output experiments. Typically, these experiments consist of raining on the surface of a soil and measuring the drainage... 

Nonautonomous time-periodic forcings with frequency cui can be treated analogously. Indeed, to = uj t) and a = cr t) then depend on time t directly and our remarks on (3.20), (3.21) apply. We will return to this observation when we describe the alternative kinematic approach, in section 3.3. 

The kinematic approach to the propagation of planar excitation waves idealizes the wave front location to be given by a one-dimensional curve... 

V. Perez-Munuzuri, M. Gomez-Gesteira, and V. Perez-Villar. A geometrical-kinematical approach to spiral wave formation super-spiral waves. Physica D Nonlinear Phenomena, 64 420-430, 1993. 

Strain-Rate Estimates in the Stir Zone. Strain rates during FSW have not been measured experimentally. However, several modeling techniques have been used to estimate the strain rates during FSW of aluminum alloys, including a kinematic approach (Ref 8), CTH or hydrocode (Sandia National Laboratories) (Ref 9), computational fluid dynamics models (Ref... 

Tolerancing (Kinematic Approach to). Table 1 Common tolerance zone shapes and associated torsor parameters Tolerance zone and typical... 

Tolerancing (Kinematic Approach to). Fig. 1 SDTI for a plane on a part with dimension d, dimensional tolerance t, and parallelism p, p < t... 

Soubra, A. H., Macuh, B. (2002). Active and passive earth pressure coefficients by a kinematical approach. In Proceedings ofinstitution of Civil Engineers. Geotechnical Engineering, 155(2), 119-131. 

Modelling and analysis of composites thermoforming KINEMATIC APPROACH... 

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