Approximation kinematic

In the concepts developed above, we have used the kinematic approximation, which is valid for weak diffraction intensities arising from imperfect crystals. For perfect crystals (available thanks to the semiconductor industry), the diffraction intensities are large, and this approximation becomes inadequate. Thus, the dynamical theory must be used. In perfect crystals the incident X rays undergo multiple reflections from atomic planes and the dynamical theory accounts for the interference between these reflections. The attenuation in the crystal is no longer given by absorption (e.g., p) but is determined by the way in which the multiple reflections interfere. When the diffraction conditions are satisfied, the diffracted intensity ft-om perfect crystals is essentially the same as the incident intensity. The diffraction peak widths depend on 26 m and Fjjj and are extremely small (less than... 

For particles of heavy atoms such as Au or Pt it is not sufficient to assume that the calculations of diffraction patterns can be made by use of the simple, single-scattering, kinematical approximation. This leads to results which are wrong to a qualitatively obvious extent (16). The calculations must be made using the full dynamical diffraction theory with the periodic... 

Table 1. Comparison of structure factor modulii Vg for tetracontane based on the known structure for hkO reflections in the [001] zone axis. Four different kinematic approximations are used for reflections whose observed intensity exceeds three times the background. The R-factor indicates the fit of different approximations.

Depending on the type of the DP there exist different formulas for the transmission case which relate the intensity with the structure amplitude in the kinematical approximation [1,2], The key-formulas for integral intensity are ... 

The most important question for the calculation of the structure amplitudes from the intensities is that for the validity of the kinematical approximation. Due to the strong interaction of fast electrons with matter the effects of dynamical scattering become more pronounced with increasing size of the microcrystallites in the film. In order to justify application of the kinematical equations it is necessary that the diffracted intensity is much less... 

Large differences between experiment and theory are often the indication of systematic errors, such, deficiency in theoretical model (as in the case kinematics approximation for dynamically scattered electrons) an measurement artifacts (such as uncorrected distortions). 

When all the phases present were identified, we can quantify their volume fraction in the analyzed volume similarly to the way the Rietveld-method is used for phase analysis in XRD. A whole profile fitting is used in ProcessDifraction, modeling background and peak-shapes, and fitting the shape parameters, thermal parameters and volume fractions. Since the kinematic approximation is used for calculating the electron diffraction intensities, the grain size of both phases should be below 10 nm (as a rule of... 

As a stmcture becomes more complex and the number of unique atoms increases, phases derived by direct methods become less reliable, especially when the electron diffraction data deviate from the kinematical approximation because of dynamic effects. HREM combined with crystallographic image processing provides a unique method for determining such stmctures. HREM images from a number of projections along different zone axes may be combined into a 3D potential map. 

The number of satellite peaks will depend on the shape of the interface between the units. It is convenient to think of the diffraction pattern in the kinematic approximation as the Fourier transform of the structure. If the layers in the units were graded so that the overall structure factor variation were sinusoidal, this would have ordy one Fourier component and thus only one pair of satellites. If the interface is abrapt, this is equivalent to the Fourier transform of a square wave, which consists of an infinite number of odd harmonics the corresponding diffraction pattern is also an infinite number of odd satellites. The intensities of the satellites therefore contain information about the interface sharpness and grading. 

Most of the applications of electron diffraction intensities for structure analysis rely on a kinematical approximation and thus do not account for the effects of dynamical multiple diffraction. The use of intensities which may be strongly perturbed by multiple scattering results in many cases in poor or misleading structure indications in the direct methods results. One approach which can be shown to reduce dynamical effects somewhat is to use precession electron diffraction (RED) [67] which involves conical rotation of the incident beam about a zone axis direction and thus avoids the strongly dynamical direct zone axis orientation. Although the intensities collected with this technique are still significantly perturbed by dynamical effects [68, 69] results obtained by this approach for zeoHtes are encouraging [70-72]. 

As has become clear in previous sections, atomic thermal parameters refined from X-ray or neutron diffraction data contain information on the thermodynamics of a crystal, because they depend on the atom dynamics. However, as diffracted intensities (in kinematic approximation) provide magnitudes of structure factors, but not their phases, so atomic displacement parameters provide the mean amplitudes of atomic motion but not the phase of atomic displacement (i.e., the relative motion of atoms). This means that vibrational frequencies are not directly available from a model where Uij parameters are refined. However, Biirgi demonstrated [111] that such information is in fact available from sets of (7,yS refined on the same molecular crystals at different temperatures. 

The refinement (2) proceeded in the same way as for the x-ray work, except that o and S q were refined as additional variables. We assumed that the scattering was kinematic. The cross sectional dimensions of the microfibrils are 200xl0C)X and our previous work on synthetic polymer single crystals showed that the kinematic approximation was adequate for such small crystallites. Intensity measurement presented considerable difficulty in that multiple film exposures could not be obtained. Sequential exposures of the same area of the specimen led to problems of beam damage, and patterns from different areas were not comparable due to differences in the preferred orientation. As a result, only the 28 strongest non-meridional intensities could be measured. These were all for reflections which could be indexed by the Meyer and Misch unit cell, and thus the two chain unit cell was used for the refinement. 

The second method uses the kinematic approximation in which the reflectivity is given as [21]... 

To date, the matrix method has been the preferred method applied for the analysis of neutron reflectivity data from the solid-solution interface, but in many cases, the kinematic approximation offers more flexibility. This has been exploited in studies at the air-solution interface, where both approaches are extensively used. 

The diffraction intensity of ultrathin films and small nanocrystals can be approximated by sununing scattering from an assembly of atoms (kinematic approximation) ... 

The kinematic approximation breaks down at a certain crystal thickness when the diffracted intensity approaches that of the incident beam. A usefiil criterion for kinematic approximation is r < fg/4, where fg is the extinction distance of the strongest reflection in the diffraction pattern. The extinction distance is orientation dependent. In case only one set of lattice planes is strongly diffracting (the two-beam condition), the extinction distance is given by = h lmeX Vg. 

Instead of using the optical matrix method of analysis, a more direct insight into the structure of the spread monolayer can be obtained using the kinematic approximation (Simister et al., 1992 Lu et al., 1992). The kinematic approximation is valid for Q Q. and for low values of reflectivity [ (Q) < 10" ], and in these cases the specular reflectivity can be written as... 

Since the interaction of hard X-rays with matter is small, the kinematical approximation of single scattering is valid in most cases, except for perfect crystals near Bragg scattering. The intensity scattered by a block-shaped crystal with N, q and N, unit cells along the three crystal axes defined by the vectors Uj, a and a, takes the form ... 

In the technique of spot-profile analysis LEED (SPA-LEED) [29, 30] intensity variations across LEED diffraction spots are measured. The technique provides information on both periodic and non-periodic arrangements of superstructure domains, terraces or facets, and strained regions. The interpretation of spot profiles is simplified due to the validity of the kinematic approximation, which is not the case for LEED I-V analysis. Deviations from simple structures resulting from defects produce characteristic modifications of the spot profile, so that a quantitative evaluation is possible. SPA-LEED has proven itself to be useful in the study of dynamic phenomena on surfaces such as phase transitions. Several examples of this capability are discussed in section 3 for Pb on Cu. 

This equation is commonly referred to as the kinematic expression and is valid whenever the perturbation of the incident free space wavefunction by the scattering medium is sufficiently small. A highly reflective scattering event greatly perturbs the incident wavefunction and consequently the kinematic approximation is inadequate for such interactions. 

To evaluate the adequacy of the kinematic approximation one can compare the reflectivity as calculated via Eq. (3.34) and compare it to the output from Eq. (3.21) for a simple SLD profile. The SLD profile for a smooth interface between air and quartz is shown in Figure 3.3a where the function p(z) is seen to be a Heaviside function... 

Heaviside function, (b) Comparison of reflectivity curves for the SLD profile using the matrix method (solid curve) and the kinematic approximation (dotted curve). 

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]... 

In an electrochemistry-NR experiment, the reflection of neutrons takes place at an interface consisting of five parallel phases as schematically shown in Figure 3.9. Table 3.1 lists the numerical values of the theoretical SLDs of the materials used in typical electrochemical studies. Each phase contributes to the overall measured NR, and to understand the shape of the experimental reflectivity curves it is instructive to examine the contribution of each individual lamina. To do this, a recursion scheme for stratified media described by Parratt [18] can be used to calculate the reflectivity of a simulated interface. These calculated reflectivities are then compared to the reflectivities predicted by the kinematic approximation. Consider a 20 A thick film of a hydrocarbon-based surfactant deposited on a gold/ cliromiurn-modified quartz sample. To simplify the analysis, a mixed D2O/H2O... 

In 1912, when M. Lane suggested to W. Friedrich and P. Knipping the irradiation of a crystal with an X-ray beam in order to see if the interaction between this beam and the internal atomic arrangement of the crystal could lead to interferences, it was mainly meant to prove the undulatoiy character of this X-ray discovered by W.C. Rontgen 17 years earher. The experiment was a success, and in 1914 M. Laue received the Nobel Prize for Physics for the discovery of X-ray diffraction by crystals. In 1916, this phenomenon was used for the first time to study the structure of polycrystalhne samples. Throughout the 20 century. X-ray diffraction was, on the one hand, studied as a physical phenomenon arrd explained in its kinematic approximation or in the more general context of the dynamic theory, and on the other, implemented to study material that is mainly solid. 

Equation [21] shows water transport as a diffusion-convection process. If the diffusive term V.(/)V0) is neglected, Eq. 121 ] reduces to Eq. IK with r(0,r) = 0A7r)O), . This type of kinematic approximation to wilier transport has been used... 

We have used a lattice-gas model to analyze preferential flow mechanisms in porous media. This numerical approach provides information that most experimental methods are presently unable to supply. The need for simple macroscopic models of preferential flow oriented us to test the kinematic wave approach. The comparison of the lattice gas solution with the kinematic approximation has shown the limitations of the latter approach and the possible ways of ameliorating it. 

In the kinematic approximation the intensity of scattering, as discussed in Section 1.5.2, is given by... 

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