Bragg component

The initial stages of powder diffraction structure analysis often require the decomposition of the powder pattern into individual Bragg components without reference to a structural model. The DDM-based decomposition procedure consists in finding additions to the calculated (or initially set) reflection intensities for minimizing the squared derivatives of the difference... 

Figure 3.4.2.34 Model of surface roughness of the sharp Bragg component is constant, involving only two levels. The characteristic but the diffuse component has a width that island size corresponds to a specific corre- depends on the correlation length short for lation length and leads to a two-component the system on the left and long for the one line shape. For a fixed coverage 9, the height on the right.

For the two-level system, we have the sum of four terms in Eq. (3.4.2.82), which can be grouped into terms with and terms without the correlation function c r). The terms without c(r) correspond to a sharp peak component that we call the Bragg component, Ib, the terms with c(r) are broad and are called the diffuse component. Id- The components are given by... 

The total peak profile is the sum of these two components, with a relative weight that depends on 0 and I. The Bragg component is mathematically a 5-function, but is in reality broadened by instrumental and sample imperfections. The shape of the diffuse component depends on the form of the correlation function. Many forms are possible, but two common ones are an exponential correlation function, leading to a Lorentzian line shape, and a Gaussian correlation function, leading to a Gaussian Hne shape. One can also use correlation functions that describe a preferred distance (e.g., island-island correlations) or with an in-plane anisotropy [42]. 

Figure 3.4.2.35 A crystal truncation rod for amplitude, does not depend on the rough-the two-level model with 0 = 0.3. Along the ness if both components are integrated. In rod, the relative weight of the Bragg and practise, often only the Bragg component diffuse components changes. The total inte- can be measured, leading to a loss in inten-grated intensity, and thus the structure factor sity in the tails of the rod.

The relative weight of the two components depends on I (see Figure 3.4.2.35). Close to the bulk Bragg peaks, the sharp Bragg component dominates near I = 0.5, the relative contribution of the diffuse component is maximal. For the simple cubic two-level system discussed here, the absolute value of the diSuse contribution is in fact constant because the product of and (1 — coslnl)... 

We can now distinguish two cases integer and fractional-order reflections. For the h integer, only the Bragg component is present because the phase factor in the diffuse component makes this contribution zero. For h half-order, on the other hand, we see that the Bragg component disappears and only the diffuse component remains. Thus... 

This result can be generalized to a multistate system. If 9j is the fraction of the surface with structure factor Fj, the integrated intensity of the Bragg component is 2... 

If, on the other hand, the domain size is so large that the diffuse component is included in the integration, the modeling of the surface requires an incoherent addition. As an extreme example, consider a surface divided in two halves, each with a different structure factor. The domains are so large that the diffuse component is indistinguishable from the (instrument broadened) Bragg component. The intensity is therefore the incoherent, weighted sum of the separate halves, exactly as expected for two independent parts. 

In case the difiuse and Bragg components can be clearly distinguished, one can choose either coherent or incoherent summation, or even use both in a structure refinement [45]. 

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