Diffraction kinematical theory

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. 

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 ... 

For particles of heavy atoms such as Au or Pt it is not sufficient to assume that the calculations of diffraction patterns can be made by use of the simple, single-scattering, kinematical approximation. This leads to results which are wrong to a qualitatively obvious extent (16). The calculations must be made using the full dynamical diffraction theory with the periodic... 

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... 

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. 

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. 

G.26 Leonid V. Azaroff. Elements of X-Ray Crystallography (New York McGraw-Hill, 1968). Crystallography, diffraction theory (kinematic and dynamic), structure analysis, single-crystal and powder methods. 

The reason lies in the non-applicability of the kinematical approximation (theory of single scattering) in the case of diffraction of electrons by single crystals. This, in comparison with X-ray quanta, is due to the much stronger scattering of the electrons at the lattice atoms and the consequent very high intensity of the singly and multiply diffracted electrons. 

To determine the structural parameters of these M-plane MQWs by HRXRD, we follow the same procedure as employed in Ref. 67. Symmetric X-ray o)-20 scans were taken with a Philips X Pert PRO triple-axis diffractometer with a GuXai source, a Ge(220) hybrid monochromator, and a Ge(220) three-bounce analyzer crystal. We first analyzed the angular positions of the satellite peaks kinematically based on the nominal growth times to obtain the structural parameters of the sample, implicitly assuming that segregation does not occur. Next, we employed simulations based on the dynamical diffraction theory [67] and varied the kinematically obtained parameters, until the intensity of the satellites matched the experiment in addition to their position. The simulations were performed for perfect interfaces and are convoluted with the instrumental resolution (25 ) only. As an example. Figure 6.15 shows (o-20 scans for sample MQW-A across the (a) GaN(nOO) and (h) GaN(2200) reflection. 

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. 

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... 

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