Weak beam

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],... 

Single-Particle Diffraction, Weak-Beam Dark-Field, and Topographic Images of Small Metallic Particles in Supported Catalysts... 

Figure 7. Weak beam images of a)octahedral and b)cubo-octahedral particles.

Figure 8. Platelet structures observed by weak beam dark field.

The new TEM techniques can provide a full characterization of small particles. The combination of weak beam images and microdiffraction information can render a very complete picture of the particle structure. In addition, refracted electron images can be... 

This effect might be interpreted by the Bethe dynamic potential approximation, which does not take into account the crystal orientation (as in the Blackman correction case) nor crystal thickness parameters. In terms of this approach, the effect of weak beams can be included in two-beam theory by replacing the potential coefficients, Vh, by ... 

Here indexes h and g denote the three indexes hkl. The interaction between weak beams is ignored. The approximation enables to estimate the effect of a weak wave on the field of the strong waves and is sufficient for taking into account the systematic interactions within a series of reflections containing one strong reflection ho. 

Here the summation is over all weak beams h. The Bethe potential accounts for the perturbational effects of weak beams on the strong beams. The criterion for a weak beam we use is... 

A further important consequence of the small source size and low divergence is that it now becomes possible to undertake weak beam topography. Weak beam... 

Figure 10.15 Simulated image width as a function of deviation parameter in Bragg case weak beam topographs. Here, the specimen is set off the Bragg peak and an image of the defect occurs only when the lattice planes are locally rotated or dilated back into the Bragg condition. As this occurs only close to the dislocation core, the images are narrowed from those under strong beam conditions...

Pt-Ir Coimpregnating on A1203. / - / - High resolution weak beam images. 

Fig. 7.7. TEM images and SAD patterns (insets) of a poly crystalline ZnO film on silicon (111) PLD grown at 1 x 10 3mbar O2 and about 540°C (a) Bright field Si(lll) plane view observation, grain size is about 70 nm, (b) cross-section HRTEM lattice image with intermediate SiO layer, and (c) weak beam Si(110) TEM cross-section. The area from which the SAD patterns were taken are within the white circles. Reprinted with permission from [49]...

The TEM investigations were performed in a 200 keV Philips CM 200 FEG/ST microscope which is equipped with a field emission gun. Dislocation Burgers vectors b were analyzed on the basis of the b g = 0 extinction criterion using different imaging vectors g. The weak-beam... 

Figure 1. Cross-section weak-beam images of the same area of a MOVPE-grown ZnO layer on Al203(0001) with a 40 nm GaN buffer layer taken with (a) g = 0002 under g,3g conditions and (b) g = 11-20 under g,3g conditions.

This equation is particularly useful when setting up for weak-beam dark field imaging, to be discussed in Chapter S. 

Establishing kinematical BE and weak beam DF conditions. The procedures to be followed for establishing these diffracting conditions are best described in terms of the Ewald sphere construction (Sections 3.4 and 3.7). 

Figure 5.19. Ewald sphere diagrams and corresponding diffraction patterns illustrating the procedures for setting up the conditions for weak beam dark field imaging using the first-order diffracted beam g. Continued, p. 160)...

Cockayne (1973) has given a detailed account of the principles and practice of the WBDF technique, including weak beam image profiles of dislocations calculated using a six-beam dynamical theory. The diffracting conditions that optimize the capabilities of the microscope are the following ... 

Cockayne, D. J. H. (1973). The principles and practice of the weak-beam method of electron microscopy. J. Microscopy, 98. 116-34. 

Ishida, Y., Ishida, H., Kohra, K., Ichinose, H. (1980). Determination of the Burgers vector of a dislocation by weak-beam imaging in a HVEM. Phil. Mag., A42, 453-62. 

---