Kinematic theory of diffraction

We know that each atom scatters the incident radiation and that the amplitude of the scattered wave, called the atomic scattering factor, depends on the number of electrons in the atom in qnestion. We are now going to make two assumptions  

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. 

Diffracted amplitude structure factor andform factor 

First of all, consider a crystal cell, the origin of which is located at an arbitrary point M of the crystal lattice. Inside this cell, an arbitrary point N can be defined by 

If this cell is irradiated with a parallel beam of X-rays traveling along the direction of the vector Sq, then by applying the same process as before, it is possible to calcnlate the amplitude scattered by this cell. This amplitude is written  

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Image contrast in terms of the kinematical theory of diffraction... 

In developing the kinematical theory of diffraction in Chapter 3, we assumed that a beam of electrons has wavelike characteristics and can be described by an equation of the form... 

Hirsch, P. B., Howie, A., Whelan, M. J. (1960). A kinematical theory of diffraction contrast of electron transmission microscope images of dislocations and other defects. Phil. Trans. Roy. Soc. (London), A252, 499-529. 

Diffraction by an ideal mosaic crystal is best described by a kinematical theory of diffraction, whereas diffraction by an ideal crystal is dynamical and can be described by a much more complex theory of dynamical diffraction. The latter is used in electron diffraction, where kinematical theory does not apply. X-ray diffraction by an ideal mosaic crystal is kinematical, and therefore, this relatively simple theory is used in this book. The word "mosaie" describes a crystal that consists of many small, ideally ordered blocks, which are slightly misaligned with respect to one another. "Ideal mosaic" means that all blocks have the same size and degree of misalignment with respect to other mosaic blocks. Most of this chapter is dedicated to conventional crystallographic symmetry, where three-dimensional periodicity is implicitly assumed. 

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. 

The second chapter is dedicated to properties and sources of radiation suitable for powder diffraction analysis, and gives an overview of the kinematical theory of diffraction along with its consequences in structure determination. Here, readers learn that the diffraction pattern of a crystal is a transformation of an ordered atomic structure into a reciprocal space rather than a direct image of the former. Diffraction from crystalline matter, specifically from polycrystalline materials is described as a function of crystal symmetry, atomic structure and conditions of the experiment. The chapter ends with a general introduction to numerical techniques enabling the restoration of the three-dimensional distribution of atoms in a lattice by the transformation of the diffraction pattern back into direct space. 

This section will enable us to return to the Laue conditions which we showed with the help of the kinematic theory of diffraction. These conditions express the fact that the waves scattered by each of the points on a row have to be in phase. 

X-ray photon is scattered only once. Scattering of diffracted beams back into the incident beam direction is ignored. This reasonable approximation is the basis of the kinematical theory of diffraction, which underpins the theoretical calculation of the intensity of a diffracted beam of X-rays. 

This case actually corresponds to the conditions where the kinematic theory of diffraction is applied, and, in the terms of the parameters here introduced, to the condition A I- For this condition, both relations... 

The key here was the theory. The pioneers familiarity with both the kinematic and the dynamic theory of diffraction and with the real structure of real crystals (the subject-matter of Lai s review cited in Section 4.2.4) enabled them to work out, by degrees, how to get good contrast for dislocations of various kinds and, later, other defects such as stacking-faults. Several other physicists who have since become well known, such as A. Kelly and J. Menter, were also involved Hirsch goes to considerable pains in his 1986 paper to attribute credit to all those who played a major part. 

G.43 Leonid V. Azaroff Roy Kaplow N. Kato Richard J. Weiss A. J. C. Wilson and R. A. Young. X-Ray Diffraction (New York McGraw-Hill, 1974). Advanced treatments of atomic scattering, kinematical and dynamical theories of diffraction, powder diffractometry, and single-crystal intensities. 

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