Kinematical theory

RHEED intensities cannot be explained using the kinematic theory. Dynamical scattering models of RHEED intensities are being developed. With them one will be able to obtain positions of the surface atoms within the surface unit cell. At this writing, such modeling has been done primarily for LEED. 

Kinematic Theory of Multiple-Order Reactions Involving Stripping or Complex Formation... 

Figure 3. Variation of the relative primary, secondary, and tertiary ion currents with ionization chamber concentration as predicted by the kinematic theory for the three models of complex formation, hydrogen ion, and hydrogen atom stripping. For conditions, see text...

The occurrence of multiple-order reactions in the methanol system is shown by the increase of R with N in Figure 2, and the system can be analyzed in terms of the kinematic theory of hydrogen transfer developed above. Consider the situation for CH3OD for which the following reaction sequence applies. 

The kinematic theory can now be used to determine the appropriate values of Js, lQtotal, and It (Equations 15, 16, and 17, respectively) for various assumed values of k, the rate constant of Reaction O—a quantity about which nothing is known—and for the various mechanisms. Obviously from the chemistry of the system, Reactions U, V, and W cannot occur by a hydrogen atom transfer mechanism hence, only two cases need be considered—Reactions S, T, U, V, and W, occurring by... 

Figure 5. Comparison between the experimental variations of R, the ratio CH3OD2 V CHjOHD +, with ionization chamber concentration of CHsOD and theoretical predictions of the kinematic theory for assumed velocity-independent rate constants of the reaction CtUOH2 + + CH5OH - CH3OH + CH3OH2+ for both the complex-formation and proton-stripping mechanisms...

The prindple of a LEED experiment is shown schematically in Fig. 4.26. The primary electron beam impinges on a crystal with a unit cell described by vectors ai and Uj. The (00) beam is reflected direcdy back into the electron gun and can not be observed unless the crystal is tilted. The LEED image is congruent with the reciprocal lattice described by two vectors, and 02". The kinematic theory of scattering relates the redprocal lattice vectors to the real-space lattice through the following relations... 

It is conventional and useful to approach X-ray scattering theory on two levels, the so-called kinematical and dynamical theories. The simpler kinematical theory assumes that a negligible amormt of energy is transferred to the diffracted beam, with the cotrsequence that we can ignore rediflfaction... 

We began the discussion of kinematical theory in Chapter 1, showing how the scattering from atoms is added up with regard to the phase to form the scattering from a unit cell, the structure factor. We repeat this important eqrration here. The structure factor for the hkl reflection is... 

The best equations to use for intensity are those of the dynamical theory, but kinematical theory gives some useful insights, based simply upon the idea that the scattering increases monotonically with the stractrrre factor, equation (4.1). 

This is the most useful quantitative intensity formula that may be derived from kinematical theory, since it is applicable to thin layers and mosaic blocks. We add up the scattering from each unit cell in the same way that we added up the scattering from each atom to obtain the stractme factor, or the scattering power of the unit cell. That is, we make allowance for the phase difference r, . Q between waves scattered from unit cells located at different vectors ri from the origin. Quantitatively, this results in an interference function J, describing the interference of waves scattered from all the unit cells in the crystal, where... 

For large single crystals extremely narrow rocking curves are predicted by the kinematical theory and these are not found. Dynamical theory is required for these cases. 

Here is seen the most obvious contrast with the kinematical theory. Both curves are plotted as a function of thickness in Figure 4.23. It is seen that the formulae give similar results for small thicknesses, that the kinematic theory diverges drastically at larger thicknesses, and that the dynamic intensity shows oscillations about a saturated level after a thickness of about. ... 

From kinematic theory, the electron density distribution in a crystal may be represented by a Fourier series,... 

We now consider the diffraction of electrons by a single crystal in terms of the so-called kinematical theory. Although this theory has serious limitations, it is useful in practice under certain conditions, and it also provides an introduction to the more satisfactory dynamical theory, which we develop in Chapter 4. 

In the kinematical theory, we consider the diffraction of a plane wave (of wavelength X) incident upon a three-dimensional lattice array of identical scattering points, each of which consists of a group of atoms and acts as the center of a spherical scattered wave. Our problem is to find the combined effect of the scattered waves at a point outside the crystal, at a distance from the crystal that is large compared with its linear dimensions. In developing the theory, we make several important assumptions ... 

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