Bragg peak interference

Fundamentals. The scattered X-ray from crystal planes or a thin metal film can interfere coherently with scattered X-rays from an adsorbate layer. This phenomenon is called surface differential X-ray diffraction (SDD) or Bragg peak interference. The theory has been described elsewhere in detail [43,44], Information on the relative amount of material in the adsorbate layer (coverage) and the distance between atoms in the top layer of the substrate and in the adsorbate layer as a function of electrode potential can be derived from SDD measurements. 

Figure 1 Bragg diffraction. A reflected neutron wavefront (D, Dj) making an angle 6 wKh planes of atoms will show constructive interference (a Bragg peak maxima) whan the difference in path length between Df and (2CT) equals an integral number of wavelengths X. From the construction, XB = d sin 6.

The intensity variation along CTR s is particularly sensitive to the difference between the bulk and surface structures. Let us take for instance a surface whose last interplanar distance is b instead of a. This produces a large asymmetry of the CTR intensity around Bragg peaks. The larger the maximum value of the perpendicular momentum transfer, the larger the interference term and thus the asymmetry, and the better the accuracy on this relaxation. Hence, the measurement of CTR s allows the determination of the atomic structure of a surface. 

This inverse Fourier transform calculation of the correlations of density of scattering centres of the sample gives particularly precise results when this sample is a crystal. In this case p(f) is periodic. The scattered intensities are then 8 functions , or Dirac s functions, that are zero almost everywhere, except for well-defined values of 2 where they take on great amplitudes. They are known as Bragg peaks for which all scattered waves have the same phase. Interferences of all these waves are consequently constructive in the directions where Bragg s peaks appear. This is the consequence of the mathematical result that the... 

Interference function scattering was modelled by using a paracrystalline lattice model as a basis, with the adoption of the equations for X-ray scattering from low molecular weight crystalline materials. Thus the Bragg peaks essentially result from a powder diffraction pattern from the block copolymers. Reasonable agreement... 

For these reasons, it may be said that the use of the purety repulsive potential in the correlation hole theory is not justified. The correlation hofe is a theoretical concept based on such a potential. Furthermore tlrc tlooietical outcome is inconsistent with observation. Therefore, such a statement by Halte, Wenner-strom and Piculell [86] based on the correlation hole ccmcept that our interpretation (in terms of the spacial ordering) needs revision is dearly unwarranted Similarly, the statements by Kaji et al. that the correlation hole concept was supported both by theory and experiments [87] and that tlo idea of identifying the observed interference peak as the Bragg peak may be negated [35] are spurious. It seems that these authors idoitify the gravy as the goose. 

When there is constructive interference from X rays scattered by the atomic planes in a crystal, a diffraction peak is observed. The condition for constructive interference from planes with spacing dhkl is given by Bragg s law. 

Fig. 9. X-ray intensity distributions (arbitrary scale) from aggregates formed by different polyglutamine peptides (Q , for n = 8,15, 28, 45) polyGln45 (dried), polyGln28 (vapor hydrated), polyGln15 (vapor hydrated), and polyGlng (lyophilized). The vertical bars indicate the positions of the Bragg reflections. The first interference peak for slab stacking of Q8 is indicated by. See Sharma et al. (2005) for further details.

When the epitaxial layer thickness is quite high, typically of the order of one micrometre, we can apply the simple criteria discussed in Chapter 3 to determine the layer parameters from the rocking curve. The effective mismatch can be determined by direct measurement of the angular splitting of the substrate and layer peaks and the differential of the Bragg law. This simple analysis catmot be applied when the layer becomes thin, typically less than about 0.25 //m, where, even for a single layer, interference effects become extremely important. We consider these issues in section 6.2 below. 

Powder x-ray spectroscopy can employ smaller crystalline samples from 1 to several hundred nanometers. These crystallites have broadened peak profiles as a result of incomplete destructive interference at angles near the Bragg angle defined as... 

As the crystallite size decreases, the width of the diffraction peak increases. To either side of the Bragg angle, the diffracted beam will destructively interfere and we expect to see a sharp peak. However, the destructive interference is the resultant of the summation of all the diffracted beams, and close to the Bragg angle it takes diffraction from very many planes to produce complete destructive interference. In small crystallites not enough planes exist to produce complete destructive interference, and so we see a broadened peak. 

To get constructive interference, we have to fulfill the Bragg condition. Inserting Eq. (A.l) into Eq. (A.6) leads to q = 2it/d. In other words we observe a diffraction peak, if the scattering vector is perpendicular to any of the lattice planes and its norm is equal to... 

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