Kinematical diffraction

In a difiraction experiment one observes the location and shapes of the diffracted beams (the diffraction pattern), which can be related to the real-space structure using kinematic diffraction theory. Here, the theory is summarized as a set of rules relating the symmetry and the separation of diffracted beams to the symmetry and separation of the scatterers. 

For a review on the formulation of kinematic diffraction theory with emphasis on the scattering of low-energy electrons, see M. G. Lagally and M. B. Webb. In Solid State Physics. (H. Ehrenreich, E Seitz, and D. Turn-bull, eds.) Academic, New York, 1973, Volume 28. 

Assuming kinematical diffraction theory to be applicable to the weakly scattering CNTs, the diffraction space of SWCNT can be obtained in closed analytical form by the direct stepwise summation of the complex amplitudes of the scattered waves extended to all seattering centres, taking the phase differenees due to position into aeeount. 

The pseudo-WPOA theory proves the validity of introducing diffraction crystallographic methods based on the kinematical diffraction theory into HREM stmcture analysis. 

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. 

To allow a comparison between dynamic and kinematic diffraction the MSLS program was also used to simulate a kinematic refinement. In order to approximate a kinematic refinement with the MSLS program, a very small thickness (i.e. 1 nm) or a very low occupancy should be taken. The latter approach was used because in this way the thickness and orientation dependence of the shape of the diffraction spots is properly taken into account. In the calculation of the kinematic R values a 0.1% occupancy of all atom sites and the thicknesses obtained for the dynamic refinement were taken. 

The R values assuming kinematic diffraction are also given in Table 4. For the calculation of these R values the MSLS program was used with occupancies of 0.1% only the scale factors were refined. Evidently these R values are much higher then the ones taking into account the d5mamic scattering. 

The first point is one of simplicity, as the integrated intensity varies hnearly with thickness when the layer is sufficiently thin to be in the kinematical diffraction limit. This is again valid irrespective of whether the layer is imperfect or perfect. It may be more convenient to measure the ratio of layer to substrate peak, which also varies linearly with thickness provided that the layer remains sufficiently thin for absorption to be neglected (Figure 6.2). 

The constant Ce = 7.29 10 is determined by the electron energy which was measured precisely for our electron microscope by convergent beam electron diffraction (Kmse et. al., 2003). Equation (2) only applies if kinematical diffraction conditions are chosen. To eliminate the influence of d5mamical electron diffraction, the sample was tilted out of the zone axis by minimizing the excitation of the Bragg reflections in the diffraction pattern. 

The diameter of the tube is determined from the equatorial oscillation, while the chiral angle is determined by measuring the distances from the diffraction lines to the equatorial line. The details are as follows. The diffraction of SWNT is well described by kinematic diffraction theory (Section 3). The equatorial oscillation in the Fourier transformation of a helical structure like SWNT is a Bessel function with n = O which gives ... 

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. 

Kinematical diffraction Diffraction theory in which it is assumed that the incident beam only undergoes simple diffraction on its passage through the crystal. No further diffraction occurs that would change the beam direction after the first diffraction event. This type of diffraction is assumed in most crystal structure determinations by X-ray diffraction. Kinematical theory is well applicable to highly imperfect crystals made up of small mosaic blocks. 

Mosaic blocks (mosaic spread) Tiny blocks within a crystal structure that are slightly misoriented with respect to each other. As a result of such mosaic spread, Bragg reflections have a finite width. Extinction is weaker in a mosaic crystal than in a perfect crystal, and therefore the intensities can be predicted by the rules of kinematical diffraction. 

Ehki is the extinction multiplier, which accounts for deviations from the kinematical diffraction model. In powders, these are quite small and the extinction factor is nearly always neglected. 

Below we will examine some practical applications of the theory of kinematical diffraction to solving crystal structures from powder diffraction data. When considering several rational examples in reciprocal space, we shall implicitly assume that the crystal structure of each sample is unknown and that it must be solved based solely on the information that can be obtained directly from a powder diffraction experiment and from a few other, quite basic properties of a polycrystalline material. The solution of a number of crystal structures in direct space will be based on the previously known structural data and supported by the results of powder diffraction analysis, such as unit cell dimensions and symmetry. 

Chapter six is dedicated to the solution of materials structures, i.e. here we learn how to find the distribution of atoms in the unit cell and create a complete or partial model of the crystal structure. The problem is generally far from trivial and many structure solution cases in powder diffi action remain unique. Although structure determination from powder data is not a wide open and straight highway, knowing where to enter, how to proceed, and where and when to exit is equally vital. Hence, in this chapter both direct and reciprocal space approaches and some practical applications of the theory of kinematical diffraction to solving crystal structures from powder data will be explained and broadly illustrated. Practical examples start fi-om simple, nearly transparent cases and end with quite complex inorganic structures. 

Kinematic Diffraction by Crystals 29.2.2.3.1. Lattice, Reciprocal Lattice... 

The conditions for kinematic diffraction [160] are best approximated in the weak-beam method, which consists of making a dark-field image in a weakly excited diffraction spot. The dislocation image then consists of a narrow bright line on a darker background. 

Yet, since noting that the other extreme limit (e —> oo) is applied for the treatment of the kinematic diffraction, there appears that the values of the s parameter in the field of units, although in extrapolated cases, do not contravene to the dynamical diffraction context where the presented scheme is placed. 

The decay of crystallinity is equivalent to an increase in the extinction distance, and this can result in a change from dynamic to kinematic diffraction conditions, with rise or fall of intensity. Since most polymer crystals are much thinner than an extinction distance to begin with, the result is usually a continuous decline... 

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