Laue higher order

In this section we will discuss perturbation methods suitable for high-energy electron diffraction. For simplicity, in this section we will be concerned with only periodic structures and a transmission diffraction geometry. In the context of electron diffraction theory, the perturbation method has been extensively used and developed. Applications have been made to take into account the effects of weak beams [44, 45] inelastic scattering [46] higher-order Laue zone diffraction [47] crystal structure determination [48] and crystal structure factors refinement [38, 49]. A formal mathematical expression for the first order partial derivatives of the scattering matrix has been derived by Speer et al. [50], and a formal second order perturbation theory has been developed by Peng [22,34],... 

Lewis, A.L., Villagrana, R.F. and Metherall, A.J.F. (1978) A description of electron diffraction from higher order Laue zones, Acta Cryst. A, 34, 138-140. 

A diffraction pattern with diffraction spots belonging to both zero-order Laue zone and higher order Laue zones can be used to determine the three-dimensional unit-cell of the crystal. 

In CBED, zone-axis patterns (ZAP) can be recorded near the relevant zone axis and the pattern may also include a higher-order Laue zone (referred to as a HOLZ). The radius of the first HOLZ ring G is related to the periodicity along the zone axis [c] and the electron wavelength, by = 2/kc. CBED can thus provide reciprocal space data in all three (x,y,z) dimensions, typically with a lateral resolution of a few nanometres. As in any application, corroborative evidence from other methods such as HRTEM and single-crystal x-ray diffraction, where possible, can be productive in an unambiguous structural determination of complex and defective materials such as catalysts. We illustrate some examples in later sections. 

Figure 3.12. Facing page, (a) Diagram showing the Ewald sphere cutting the reciprocal lattice rods of zero and higher order Laue zones and (b) a schematic diagram of the corresponding diffraction pattern.

The primitive cell volumes can be deduced easily from CBED unindexed patterns, as was shown by Carpenter and Page in [7]. By combining the measurements from the ZOLZ, (zero order Laue zones, which appear in the center of the CBED pattern) and the HOLZ (higher order Laue zone, which appears as rings around the ZOLZ pattern) it is possible to calculate the primitive cell... 

Figure 3 Comparison of central disks from CBED patterns (A) GaN and (B) Gao.9elno.04N. Note relative shifted positions of dark higher-order Laue zone lines due to differences in lattice parameter.

Some of the alternative TOF instrument designs involve replacing the beryllium filter with either a crystal or a mechanical chopper to monochromate the incident beam. With this change, the spectrometer can be used with a higher incident neutron energy (typically E 50 meV) so that a smaller momentum transfer Q is possible for 5 the same energy transfer (21,22). With a monochromatic incident beam, a beryllium filter is sometimes substituted for the chopper after the sample in order to increase the scattered intensity but with a sacrifice in the,minimum Q attainable. Energy transfers up to 100 meV (800 cm" ) can be achieved with TOF spectrometers at steady state reactors before the incident neutron flux is limited by the thermal spectrum of the reactor. (With hot moderators such as at the Institut Laue-... 

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