Dynamical diffraction

All tensor expressions (35)-(42) involve summation over Bloch waves, i.e. summation over j. For a dynamical diffraction calculation involving AT beams, the number of Bloch waves resulting from Equation (30) equals the number of beams, i.e. N. It should be noted, however, that not all of these Bloch waves will be strongly excited within the crystal and contribute to the electron wave field. The excitation amplitudes of the Bloch waves in the crystal are given by B 0J. Extensive numerical calculations show that in a typical dynamical diffraction calculation, although typically more than... 

Dederichs, P.H. (1972) Dynamical diffraction theory by optical potential methods, Solid State Phys., 27, 125. 

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... 

The method of strueture analysis developed by the Soviet group was based on the kinematieal approximation that ED intensity is directly related (proportional) to the square of structure factor amplitudes. The same method had also been applied by Cowley in Melbourne for solving a few structures. In 1957 Cowley and Moodie introdueed the -beam dynamical diffraction theory to the seattering of eleetrons by atoms and crystals. This theory provided the basis of multi-sliee ealeulations whieh enabled the simulation of dynamieal intensities of eleetron diffraetion patterns, and later electron microscope images. The theory showed that if dynamical scattering is signifieant, intensities of eleetron diffraetion are usually not related to strueture faetors in a simple way. Sinee that day, the fear of dynamical effects has hampered efforts to analyze struetures by eleetron diffraction. 

The measurement of ED intensities in a precession mode allows all ED intensities to be recorded under reduced influence from dynamical diffraction and secondary scattering contributions, therefore permitting structure analysis of improved precision. By processing a incident beam at a constant angle around a zone axis in combination with a similar precession of the ED pattern below the specimen the equivalent mechanism of the precession of the specimen is obtained. 

Again, it is important to note that in data taken without precession mode light atoms like lithium and oxygen do not appear well generally.Small crystal misorientations due to time of measurement and dynamical diffraction contributions to many reflections in o precession mode may well explain such results. 

Figure 5. (a) Deconvoluted image after correction of the dynamical diffraction effect (b) the same as (a) but showing the atomic arrangement schematically (c) result of deconvolution without dynamical diffraction correction and (d) simulated image based on (b). 

STRUCTURE REFINEMENT BY TAKING DYNAMICAL DIFFRACTION INTO ACCOUNT... 

Key words electron diffraction, dynamical diffraction, structure refinement... 

We have developed a software package MSLS [1], in which multi-slice calculation software is combined with least squares refinement software used in X-ray crystallography. With multi-slice calculations which are standardly used for image calculations of HREM images, dynamic diffraction is taken into account explicitly. 

Structure Refinement By Taking Dynamical Diffraction Into Account 357... 

A.AUTHIER, in X-ray and neutron dynamical diffraction theory and applications, eds. A.AUTHIER, S.LAGOMARSINO B.K.TANNER (Plenum Press, New York, 1997),p. 1. 

The dynamical diffraction theory is very accurate and highly practical, but no theoiy provides a perfect description of the complexity of real specimens. What are the limitations and approximations, and how serious are they ... 

Figure 7.8 A scattering map in reciprocal space. Equal intensity contours are shown schematically, and the Ewald sphere is represented as a plane near reciprocal lattice points 0 and h. The dynamical diffraction from the specimen is displaced slightly from the relp and from the centre of the diffuse scatter by the refractive index effect...

The second major contrast mechanism is extinction contrast. Here the distortion of the lattice arotmd a defect gives rise to a different scattering power from that of the surrotmding matrix. In all cases, it arises from a breakdown or change of the dynamical diffraction in the perfect ciystal. In classical structure analysis, the name extinction was used to describe the observation that the integrated intensity was less than that predicted by the kinematical theoiy. 

Around the defect, enhanced scattering was observed and this loss of extinction is the origin of the name. The exact nature of the images can only be explained using dynamical diffraction theory and we will return to this in a later section. 

Although systematic absences are almost always more easily recognized in electron diffraction, occasionally dynamic diffraction effects lead to a weak diffraction for a supposedly absent reflection. Repeated observations on crystals of various thicknesses, allow a decision in such doubtful cases. 

Lateral shifts are also due to depth penetration and refer to a shift in the exit location of an X-ray at the crystal surface. This results in a transverse shift at the detector, typically 10%-100% of refractive index corrections [20]. While ray-tracing can adequately describe geometrical effects outside the crystal, the effects of depth penetration and lateral shifts require full dynamical diffraction theory. 

In figure 3 a voigt fit is shown to guide the eye, while a complete lineshape model will be constructed using a code which ray-traces and solves the equations of dynamical diffraction inside the crystal [20]. Importantly, the line shows... 

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