The Kinematic Approach

The previous derivation of the reflection and transmission coefficients correctly describes the intensity of reflected neutrons at any value of momentum transfer vector. However, there is a useful alternative derivation, which gives a highly analytical function describing the reflectivity. This derivation is based on the Born approximation and is often referred to as reflectivity in the kinematic limit. Suppose there are two arbitrary but different SLD profiles pi(z) and p2 z) and one wishes to determine the separate reflectivities Ri(Q,z) and -R2(Gz) for the two scattering potentials. The solution to the problem is described by combining Eqs. (3.15) and (3.17) 

The difficulty in applying Eq. (3.33) lies in finding an appropriate expression for i//(z). An important simplification which has been widely applied in X-ray and neutron reflectivity is obtained if i//(z) is replaced with the free space wavefunction exp(iko2 ). This is known as the Born approximation and provides 

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. 

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

Substitution of Eq. (3.35) into Eq. (3.34) yields the Fourier integral for the Heaviside function H(z) 

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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. 

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. 

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... 

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

Modeling approaches commonly used can be categorized into two groups, namely, kinematic and statistical approaches. The kinematic approach uses the motion and associated forces of the conditioner and pad to describe the conditioning process. The statistical approach uses statistical models such as probability of locating a point on the pad surface for the same purpose. A classification of process models identifying key variables and major modeling objectives is presented in Table 13.4. 

Chen [61], Feng [66], Chang [67], Lee [9], Li [68], and Baisie s [69] models use kinematic approaches to predict the amount of wear across the pad by employing the Preston equation. In the kinematic approach, a major assumption is that pad wear is determined by the sliding distance on the pad. A related kinematic approach was... 

The theory of the propagation of light along the optic axis was considered in [11], [13]-[17j. The kinematical approach could explain many experimental results. However, for quantitative explanation of the experiments, a more detailed consideration is necessary. For this purpose, the dynamical theory should be used [15]. Precise measurements of the reflection spectrum from a monodomain CLC show good agreement with theory [18]. A solution... 

Until now we neglected possible interactions between the propagating waves (except for their annihilation after collisions). This is justified if the time interval between any two subsequent waves is much larger than the recovery time of the individual elements of the excitable medium. Since the rotation period of spiral waves goes to infinity in the limit Go 0 (i.e. for weakly excitable systems) and thus the spirals become very sparse there, this condition can always be satisfied close enough to the existence boundary dR of spiral waves in the parameter space defined in [4]. The kinematical approach which was formulated above allows us to find the rotation frequency of free sparse spirals and to investigate relaxation to the regime of steady rotation. 

Within the kinematic approach, the onset of meandering can be explained by simple qualitative arguments. Suppose that we have a spiral whose rotation period is not very large and therefore the wave moves through the region which has not completely recovered. It means that the effective excitability of the medium is reduced. Note, however, that inside the core region, not visited by the wave, the excitability remains unchanged. Thus the wave s... 

Kinematic or upper bound theorem If a kinematic admissible mechanism can be found for which the work of the external loads exceeds the internal work, then the stracture will collapse and the load factor (3ii) is greater than or equal to the failure load factor (5p). In the kinematic approach the failure load factor is determined by searching for the minimum load factor. 

With regard to kinematic approach, when the structure presents an enough number of hinges to transform it into a mechanism (Fig. 4b), the load factor obtained by equating the work of the external loads to zero corresponds to an upper bound limit (upper bound theorem). The load obtained from the kinematic approach is greater than or equal to the failure load. Different positions for the hinges on the structure can be adopted, and several mechanisms, and respective load factors, can be found. The mechanism that presents the lowest load factor corresponds to the collapse mechanism. Thus, the kinematic approach presents also several solutions. 

Taking into account the results obtained from both approaches, the static approach presents a safe solution of the maximum load capacity of masonry structures. However, for complex masonry structures it can be difficult to predict the position of the thrust fine. On the other hand, the kinematic approach can lead to an unsafe estimation of the ultimate load. This problem can be solved by implementing computational tools of optimizatimi and searching by the failure load (minimum load factor). 

We cannot use the equivalent force system f(x, t) and the elastodynamic Eq. 2 to determine the displacement field for a hypothetical source. The equivalent force system itself depends on the displacement field that is being sought. Two different approaches are commonly used (i) In the kinematic approach we assume some mathematically tractable displacement field in the source region (e.g., suddenly imposed slip, constant over a rectangular fault plane), derive the equivalent force system from Eq. 3, and solve Eq. 2 for the resulting displacement field outside the source region and (ii) in the inverse approach, we use Eq. 2 to determine the force system f(x, 0 from the observed displacement field and compare the result with force systems predicted theoretically for hypothesized source processes. The most useful way to parameterize the force system in this approach is to use its spatial moments. 

Figure 6.31. Calculate the minimum rotational frequency of the mandrel for the maximum layer thickness not to exceed 5 trm at the end of the die. To solve this problem follow both the geometrical and the kinematical approaches. Also, calculate the power to rotate the mandrel and introduce changes necessary for the reduction of the power consumption.

The multiscale modeling power of the kinematic approach is underscored by the PDF model prediction that filling requires a linear bidirectional spring [32], and the later experimental observation that in the early diastole heart, cells generate a restoring force driven by a linear bidirectional spring (the protein titin) [22]. 

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