Pure profile

The correct performance of any curve-resolution (CR) method depends strongly on the complexity of the multicomponent system. In particular, the ability to correctly recover dyads of pure profiles and spectra for each of the components in the system depends on the degree of overlap among the pure profiles of the different components and the specific way in which the regions of existence of these profiles (the so-called concentration or spectral windows) are distributed along the row and column directions of the data set. Manne stated the necessary conditions for correct resolution of the concentration profile and spectrum of a component in the 2 following theorems [22] ... 

Manne s resolution theorems clearly stated how the distribution of the concentration and spectral windows of the different components in a data set could affect the quality of the pure profiles recovered after data analysis [22], The correct knowledge of these windows is the cornerstone of some resolution methods, and in others where it is not essential, information derived from this knowledge can be introduced to generally improve the results obtained. 

If pure profiles can be obtained from all components, die next step in deconvolution is straightforward. 

Of the functions with three parameters IL is best suited for X-ray reflections. For Rietveld analyses mostly the four-parameter pseudo-Voigt function is used, for pure profile fitting (without structure refinement) often the Pearson-Vll function is also used. 

The profile of diffraction peaks depends on two types of contributions first, the instrumental function and, second, the stractural defects that also lead to changes in the intensity distribution. This last comment is at the core of microstructural analysis, which will be the subject of the second part of this book and will not be discussed here. Peak profiles can be described by a function h(e), where the e variable corresponds in every point to the difference with respect to the theoretical diffraction angle. The function h(e) can be expressed as the convolution product of f(e), which represents the pure profile associated with the sample s specific effects, and g(e), which constitutes the instrumental function. The function h can then be expressed as ... 

Microstractural studies require the pure profile, meaning the function f, to be determined. Solving equation [3.2] is a complex problem, dealt with later on. In this chapter, we will simply describe the different components of the instrumental function. The expressions and the relative importance of the different functions gj(e) depend on the device. Clearly, the complete description of a given diffractometer s experimental profile requires taking into account all of its elements. We will now describe the main effects observed in virtually every case. 

The essential apphcation of peak profile analysis involves the microstractural studies of the samples, which means that the machine s contribution has to be as small as possible. Digital processing techniques make it possible to properly extract the pure profile, but the quality of the study is higher if the widening of the peaks is essentially due to the sample. In practice, for this type of study, high resolution diffractometers are chosen, since their instmmental function can be very accurately determined. 

In Chapter 3, we saw that the distribution of the measured intensity h(x) can be expressed as the convolution product of the pure profile f(x) and the instrumental function g(x). We can write ... 

The problem is therefore to determine the nature and the density of the stractural defects from the measurement of the experimental profile h(x) which contains the contribution from the instrament. There are two ways to go about solving this problem. The first method consists of deconvoluting this equation by using, in particular, the properties of Fourier transforms and extracting the pure profile which induced only by the defects. The second approach is described as convolutive . This time, the stractural defects are described without extracting the pure profile, but instead by taking into account the instrument s contribution, which is assumed to be an analytical function, either known or directly calculated from the characteristics of the diffractometer. This instrumental function is then convoluted with the functions expressing the contributions from the various microstructural effects that are assumed to be present. 

The different approaches used to extract the pure profile by deconvolution were described and compared in particular by Louer and Weigel [LOU 69b], and more recently by Armstrong and Kalceft [ARM 99]. Cernansky [CER 99] gave a detailed description of the inherent mathematical aspects of this deconvolution process. Aside from Stokes historical method [STO 48], the various approaches will not be explained here, only the basic ideas will. 

The oldest method used to extract the pure profile was suggested in 1948 by Stokes [STO 48]. It consists of neglecting the experimental noise and the contribution from the continuous background, and inverting the convolution product by a Fourier analysis of the peak profile. 

This method of extracting the pure profile by Fourier analysis has seen major developments and modem computational capabilities have made it rather easy to implement. However, the calculation of the Fourier coefficients imposes that there must be no peak overlaps. This condition considerably limits the application field of this method. Additionally, as we have already mentioned, the experimental noise is assumed to be zero. The presence of a non-zero noise causes oscillations in the resulting signal after deconvolution. This problem can be solved by using the methods described below. 

In 1968, Ergun [ERG 68] suggested extracting the pure profile of X-ray diffraction peaks by using an iterative method initially described by van Citter and... 

This stabilization approach was introduced by Phillips [PHI 62] and Tikhonov [TIK 63, TIK 77], and is often referred to as the Tikhonov stabilization . It consists of a general approach to ill-posed problems, when the signal resulting from a convolution is used to try to determine the pure profile. The main idea is to limit the number of solutions to equation [6.2] by including additional information. This information, suggested by Tikhonov, is that the function is smooth or, in other words, that its second derivative is as close as possible to zero. 

Naturally, it is also possible to set more stringent conditions. One common method consists of making an assumption a priori on the shape of the functions describing the contributions from the device and from stractural defects. These functions can be defined either analytically or numerically. Usually, the relation between the measnred profile h(x) and the pure profile is expressed as a simple relation involving the integral breadths or the full widths at half maximum. 

The purpose of the process we explained in the previous sections is to extract from the experimental profile, h(x), the pure profile, f(x), which will then be used to determine the microstmctural characteristics of the sample. 

In this section, we will deal with the separation of these different contributions based on the measurement of the integral breadths and the equations we have laid out. Throughout this section, we will assume that the experimental profiles have been corrected and that the suggested methods are apphed to the pure profiles. 

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 ... 

Fig. 8.3. Autoionising resonances coupled to a power-broadened continuum. The pure profile of the power-broadened line was also determined experimentally, and is shown by the broken curve (after A. Safinya and T.F. Gallagher [401]).

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