Bragg electron microscopy

The structures of iron oxides have been determined principally by single crystal X-ray diffraction or neutron diffraction with supplementary information coming from infrared spectroscopy, electron diffraction and high resolution electron microscopy. A few years after the first successful application of X-ray diffraction to crystal structure determination, this technique was used to establish the major features of the structures of magnetite (Bragg, 1915 Nishikawa, 1915) and hematite (Bragg Bragg, 1918). 

Electron microscopy is basically a diffraction technique in which periodic crystals diffract electrons, according to Bragg s Law,... 

The total thickness d = dA + dB of a sheet is directly given by the Bragg spacing of the X-ray patterns, or directly measured on the electron micrographs (to obtain an accurate value of d by electron microscopy one must use electron micrographs provided by sections perpendicular to the planes of the sheets, this is easy with an electron microscope equipped with a goniometer head). 

In practical electron microscopy, it is often important to be able to determine accurately the deviation A0 from the exact Bragg angle 0. We now consider in some detail how we can use Kikuchi lines for doing this. 

Figure 6.23. Powder diffraction pattern collected from the NiMn02(OH) powder using Cu Ka radiation on a Scintag XDS2000 diffractometer. The experiment was carried out in a step scan mode with a step 0.02° and counting time 30 sec per step. The vertical bars indicate calculated positions of the Kai components of all possible Bragg reflections. The inset shows the scanning electron microscopy image of peuticle morphology in the as-received state.

Figure 3.15 Electron deflection by Bragg diffraction of a crystalline specimen (a) image formation in crystalline samples and (b) diffraction at crystal lattice planes and at the contours of inclusions. (Reproduced with permission from M. von Heimandahl, Electron Microscopy of Materials, Academic Press, New York. 1980 Elsevier B. V.)...

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. 

Bragg s Law In the analysis of semicrystalline polymers, in a first approach, it can be assumed that L is associated with the distance between the lamellar planes. These distances can be related through the Bragg s law with the diffracted waves that are in phase and are reinforced in certain directions (angles) [15]. This analysis give values that qualitatively coincide with the structure observed by means of electron microscopy and with sizes of crystals indicated by means of WAXS [16]. The periodicity L is related to the vector q in the maximum of dispersion according to [17]... 

However, in most cases the structural sizes inferred from this treatment do not agree with the stmctures observed by means of electron microscopy [18]. For instance, polymers crystallized from the molten state rarely have orders of multiple dispersion (as required by diffraction) and, moreover, they present a wide first-order peak as shown in Figure 19.7 [19]. Then, when applying the Bragg s law, the obtained period will be distorted considerably from the average period of the structure [14], This discrepancy of values can decrease, but not vanish, when applying the Lorentz factor to the curve of observed intensity. 

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