Bragg-diffracted beams, information

High-resolution transmission electron microscopy can be understood as a general information-transfer process. The incident electron wave, which for HRTEM is ideally a plane wave with its wave vector parallel to a zone axis of the crystal, is diffracted by the crystal and transferred to the exit plane of the specimen. The electron wave at the exit plane contains the structure information of the illuminated specimen area in both the phase and the amplitude.. This exit-plane wave is transferred, however affected by the objective lens, to the recording device. To describe this information transfer in the microscope, it is advantageous to work in Fourier space with the spatial frequency of the electron wave as the relevant variable. For a crystal, the frequency spectrum of the exit-plane wave is dominated by a few discrete values, which are given by the most strongly excited Bloch states, respectively, by the Bragg-diffracted beams. 

The intensities of the Bragg diffracted beams or reflections, 1, are determined by the atomic arrangement within the unit cells. This gives the relative positions of the atoms and thereby also information about bonding, distances and angles between the individual atoms in the crystalline compound. Mathematically this can be described as ... 

Generally the Bragg eqnation is nsefnl in describing the directions 29hki of diffracted beams, bnt provides no information to help ns to understand the magnitudes of the intensities obtained. The concepts to be described in detail in Chapter 6 are mnch more informative abont the intensities of the diffracted beams. 

The area detector - is an electronic device for measuring many diffracted intensities at one time. It is an electronic substitute for film, and is now used, where possible, for crystals of biological macromolecules. It is a position-sensitive detector, and is coupled to an electronic device for recording the data in computer-readable form. The data so recorded include the intensity of a Bragg reflection (diffracted beam) and its precise direction (as a location on the detector). Both types of information are needed for each Bragg reflection so that I(hkl), and sinO/X can be determined. 

This Chapter is concerned with methods for obtaining the relative phase angles for each Bragg reflection so that the correct electron-density map can be calculated and, from it, the correct molecular structure determined. When scattered light is recombined by a lens, as described in Chapters 3 and 6, the relationships between the phases of the various diffracted beams are preserved. In X-ray diffraction experiments, however, only the intensities of the Bragg reflections are measured, and information on the relative phases is lost. An attempt is maxle to remedy this situation by deriving relative phases by one of the methods to be described in this Chapter. Then Equation 6.3 (Chapter 6) is used to obtain the electron-density map. Peaks in this map represent atomic positions. 

Generally the Bragg s Law is useful in describing the directions, 20/, /, of diffracted beams, but it provides no information that will help us to understand the... 

The relative positions of a sufficient number of reflections arising from microstructural periodicities enable unambiguous identification of morphology. Further information can be obtained by preparing oriented specimens, and obtaining diffraction patterns for different orientations. For example, in an oriented lamellar phase with the beam incident parallel to the layers, Bragg... 

Generally the detectors are set up perpendicular to the primary beam, with the intersection of the primary beam at the detector centre. This setting has some advantages the entire Bragg cones are detected and the deviation of the cone projection from an ideal circle is usually small. Sometimes a detector can be placed off-centre and non-orthogonally to the primary beam. This can enlarge the detectable q-space in a very cost effective manner. The downsides are the strongly elliptical conical projections and the loss of the entire azimuthal information of a diffraction cone. 

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