Kinematic crystal diffraction

The simplest type of crystal diffraction, and that considered here, is called kinematical diffraction. In such diffraction the X-ray beam, once diffracted, is not further modified by additional diffraction in its passage through the crystal. The phase differences between radiation scattered at different points in the crystal depend only on differences in the path lengths of the incident and diffracted waves. The summation of these waves with the appropriate amplitudes and relative phases determines the intensities. 

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... 

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. 

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. 

Since d5Uianiical electron diffraction patterns do not obey Friedel s law ( 1(g) = I(-g) for all crystals), whereas kinematic ones do, a crystal which is known a-priori to be non-centros5mimetric in some projection, but which produces a symmetric diffraction pattern, must be diffracting under single-scattering conditions. If it was thick enough to scatter dynamically, the pattern would lack the inversion symmetry which the crystal also lacks, and so reflect the true symmetry of the crystal. 

Figure 15. Crystal structure of a-Tl2Se solved in projeetion via direct methods using quantified intensities from the selected area electron diffraction pattern shown in (a) [film data]. The potential map (E-map) in (b) was used to eonstruet an initial structural model which was later improved by kinematical least-squares (LS) refinement (c). Note that the potential of the selenium atoms in (c) appear after LS-refinement somewhat stronger than the surrounding titanium atoms (see the structural model in figure lOd). The average effective thiekness of the investigated thiekness of the crystal is about 230 A [22].

It is emphasized that the final result is the structure map of the examined crystal rather than a pseudo structure map. This is because the difftaction intensities have been pushed towards the corresponding kinematical values during the calculation of partial structure factor in each cycle of the correction. In addition, in the final step, structure refinement by Fourier synthesis modifies the peak heights towards the true values to some extent. It is obvious that all the missing structure information due to the CTF zero transfer is mended after phase extension. The amplitudes are provided by the electron diffraction data, and the phases are derived from the phase extension. As a result, the resolution of the structure analysis by this method is determined by the electron diffraction resolution limit. 

To obtain good HREM images and electron diffraction patterns is one to the most important step in 3D reconstruction. The data should be as kinematical as possible, so HREM images and electron diffraction (ED) patterns should only be taken from very thin crystals. 

Structure determination from X-ray and neutron diffraction data is a standard procedure. Starting with a rough model, the accurate structure is determined using a least-squares structure refinement, which is based on kinematic diffraction and in which the differences between calculated and experimental intensities are minimized. X-ray and neutron diffraction are not applicable to all crystals. To determine crystal structures of thin layers on a substrate or small precipitates in a matrix (see figure 1) only electron diffraction (ED) can lead you to the crystal structure. 

The diSuse scatter arises because dislocations are defects which rotate the lattice locally in either direction. This gives rise to scatter, from near-core regions, which is not travelling in quite the same direction as the diffraction from the bulk of the crystal. This adds kinematically (i.e. in intensity not amplitude) and gives a broad, shallow peak that mnst be centred on the Bragg peak of the dislocated layer or substrate since all the local rotations are centred on the lattice itself. We can model the diffuse scatter quite well by a Gaussian or a Lorentzian function of the form ... 

Figure 4.11 The kinematic dispersion surface. The circles centred on the origin O and the relp H, with radius (1/ vacuum )(1+ 2) represent, in the plane shown, the allowable wavevectors in the crystal far from the diffracting condition. A section of the Ewald sphere is shown...

Around defects, the scattering power differs from that in the perfect crystal because X-rays which do not satisly the Bragg condition in the perfect crystal may be diffracted in the deformed region arotmd the defect. Just as in the Lang projection topograph, these regions behave as small crystals which diffract kinematically and the net result is an increase in the intensity over that from the perfect crystal. 

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. 

Extinction, which is the failure of the kinematic scattering theory (Ihki hki) is only a minor problem in X-ray diffraction. In neutron diffraction, extinction is serious and pervasive throughout the whole data, as shown by the examples in Thble 3.2. The best methods available for extinction correction require careful measurement of crystal dimensions. Although somewhat empirical, it has proved to be very effective [184, 185]. At least one, and sometimes six, additional extinction parameters, gis0 or gij, have to be added to the variable parameters. Uncertainty in the validity of these extinction parameters appears to have very little effect on atomic positional coordinates, but may influence the absolute values of the atomic temperature factors. This is important in charge density or electrostatic potential... 

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. 

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 ... 

The assumptions of the kinematical theory - that the incident wave is monochromatic and plane and that there is no absorption of either the transmitted wave or the scattered waves - are reasonable assumptions with which to begin a theory. However, the other assumptions - that a scattered wave is never rescattered and that there is no interaction between the transmitted and scattered waves - are gross oversimplifications of the physical situation. For example, consider a plane monochromatic wave incident upon a crystal plate at such an angle that the Bragg law is satisfied for a set of planes approximately normal to the crystal plate, as shown in Figure 3.20. It is clear from this diagram that at A the diffracted wave Si in the crystal is rediffracted so that it travels in the same direction as the transmitted wave T. This rediffracted wave is denoted S2. There is no reason why S2 should not be also rediffracted at B to produce the wave S3, as shown. Thus, the assumption that the diffracted wave, once produced in the crystal, is never rediffracted, is clearly unacceptable. 

In developing the kinematical theory, we made use in Section 3.7 of the column approximation. In view of the preceding discussion, it is important to estimate the crystal thickness t for which the column approximation is valid. If kinematical conditions are to apply within a single column of unit cells of width x, and adjacent columns are to diffract independently of each other, then 2 t < x. Thus, for x 0.5 nm, the crystal thickness must be less than about 25 nm. 

Thus, the kinematical theory is valid only for very thin crystals that are not diffracting strongly. To interpret the details of the images observed from the much thicker crystals commonly used in TEM (particularly when s = 0), we require a theory that includes the dynamical interaction of the many beams excited in such crystals. The dynamical theory, which is developed in the next chapter, overcomes the critical limiations of the kinematical theory and provides the basis for the interpretation of the images due to crystal defects, which are discussed in detail in Chapter 5. 

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