Caglioti polynomial

This function usually increases with the diffraction angle and can often be approximated by a second degree polynomial in tangent 6. This equation is known as the Caglioti polynomial [CAG 58] and is written ... 

As we have already mentioned, one of the characteristic parameters of a peak s profile is its full width at half maximum. We saw that this width generally increases with the angle and that this evolution can, most of the time, be accurately described by a second degree polynomial in tangent 9 (the Caglioti polynomial). In a Rietveld refinement, the parameters U, V and W are refinable. Their values express both the instramental resolution function and the microstructural characteristics of the sample. We will discuss this point further in Part 2 of this book. 

In Chapter 3, we saw that the evolution of the peaks widths according to the diffraction angle can be represented by a function known as the Caglioti polynomial [CAG58] ... 

Naturally, the effects of size and microstrains are expressed as convolution products of the pure profile and the profiles resulting from each of these effects. Therefore, the modified expression of the Caglioti polynomial will depend on the hypotheses made on the shape of the profiles of each of the contributions. If we assume that the size effect leads to a Lorentzian profile, and microstrains to a Gaussian profile, we can write the following expressions ... 

The parameters U, V and W are specific to the diffractometer in question. This function was established almost 50 years ago on a neutron diffractometer. It actually has no theoretical background and its expression is essentially empirical, so for some diffractometers, the angular resolution function can stray significantly from this second degree polynomial. However, the expression established by Caglioti and his colleagues is still commonly used. 

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