Warren-Averbach analysis

As a consequence of this short discussion, we can conclude that the shapes of the distribution obtained by the Warren-Averbach analysis may be less reliable as the results can be strongly influenced by the method chosen for data reduction and treatment. On the contrary, the average crystallite dimension is very stable and it is almost independent of the adopted analytical method. 

The study of the peak shape gives important information relative to the microstructure of the sample even when it is included in the Rietveld code. Actually, in order to perform the Warren-Averbach analysis or other mi-crostructural studies, it is not necessary to use the Rietveld analysis, but it is often sufficient to operate with less complex, non-structural peak fitting procedures. 

Figure 6. Average crystallite size of TIN in ncTiN/aSi3N4 composites prepared in HF discharge [63] against the silicon content determined by the energy dispersive analysis of X-ray. Squares data from X-ray diffraction using the Scherrer formula and integral width of the Bragg reflections which were also verifled by the Warren-Averbach analysis [118,119]. Circles data from the direct lattice image in HR-TEM.

The first detailed X-ray diffraction (XRD) studies on PEMFC electrodes were performed by Wilson et al. [43] using a Warren-Averbach Fourier transformation method for determining the weighted crystallite sizes. Warren and Averbach s method takes into account not only the peak width but also the shape of the peak. This method is based on a Fourier deconvolution of the measured peaks and the instrument broadening to obtain the true diffraction profile. This method is capable of yielding both crystallite size distribution and lattice microstrain. The particle-size distributions can be determined from the actual shape of the difliaction peaks, with the use of Warren-Averbach analysis. 

Because large magnitudes of broadening were observed in PTFE, even for slowly cooled specimens, it was necessary to use line-width standards. The two materials used were annealed LiF and a diluted solid mixture of ammonium hydrogen phosphate. Data analysis proceeded by Fourier analysis of multiple orders, the well-known Warren-Averbach procedure. " Values of the domain size as measmed experimentally and with a correction using renormalized cosine coefficients (RCC), are given in Table 1.2. 

The common application of the Scherrer formula in catalyst structure determination is a crude approximation to microstructural analysis. Strain and particle size give rise to the same effects, namely, line broadening, but fortunately causing different variations with diffraction angle, as shown by Equations (3) and (4). The methodologies implied by Wil-liamson-Hall plots and Warren-Averbach profile analyses provide access to the strain and size parameters in the commonly encountered case that both phenomena contribute to an experimental line broadening. 

Direct analysis the Bertaut-Warren-Averbach method... 

A model-free method for the analysis of lattice distortions is readily established from Eq. (8.13). It is an extension of Stokes [27] method for deconvolution and has been devised by Warren and Averbach [28,29] (textbooks Warren [97], Sect. 13.4 Guinier [6], p. 241-249 Alexander [7], Chap. 7). For the application to common soft matter it is of moderate value only, because the required accuracy of beam profile measurement is rarely achievable. On the other hand, for application to advanced polymeric materials its applicability has been demonstrated [109], although the classical graphical method suffers from extensive approximations that reduce its value for the typical polymer with small crystal sizes and stronger distortions. 

Fourier transform method. The method used most widely for the separation of size and distortion in peak profiles from metals and inorganic materials is the Fourier analysis method introduced by Warren and Averbach (21). The peak profile is considered as a convolution of the size-broadening profile fg and the distortion broadening profile fj), so that the resolved and corrected profile f(x) is given by... 

In 1949, however, Warren pointed out that there was important information about the state of a cold-worked metal in the shape of its diffraction lines, and that to base conclusions only on line width was to use only part of the experimental evidence. If the observed line profiles, corrected for instrumental broadening, are expressed as Fourier series, then an analysis of the Fourier coefficients discloses both particle size and strain, without the necessity for any prior assumption as to the existence of either [9,3, G.30, G.39]. Warren and Averbach [9.4] made the first measurements of this kind, on brass filings, and many similar studies followed [9.5]. Somewhat later, Paterson [9.6] showed that the Fourier coefficients of the line profile could also disclose the presence of stacking faults caused by cold work. (In FCC metals and alloys, for example, slip on 111 planes can here and there alter the normal stacking sequence ABCABC... of these planes to the faulted... 

We have just seen how measuring these breadths enables us to quantify the defects. This analysis method implies the fitting of the peaks and therefore requires us to define a priori the shape of the diffraction peaks. Berlaut [BER 49] followed by Warren and Averbach [WAR 50, WAR 55, WAR 69] showed that with the help of a Fotrrier series decomposition of the peak profiles, any material with voltrme structrrral defects can be analyzed without having to make hypotheses on the shape of the peaks. This analysis, which we will now describe, when it can be implemented, remairts even today one of the most exterrsive ways to study microstructrtral effects based on the profiles of the diffraction peaks. 

Three different methods have been designed to quantitatively study structural volume defects. The integral breadth method, based on the theoretical considerations we discussed in Chapter 5, was introduced in 1918 by Scherrer [SCH 18] and generalized by Stokes and Wilson [STO 42], among others. Later on, Toumarie [TOU 56a, TOU 56b] followed by Wilson [WIL 62b, WIL 63] suggested a different analysis based on the variance of the intensity distribution. We described how Bertaut [BER 49] showed in 1949 that the Fourier series decomposition of the peak profile makes it possible to obtain the mean value and the distribution of the different effects that cause the increase in peak width. This method was further elaborated by Warren and Averbach [WAR 50, WAR 55, WAR 69]. 

Methods based on the Fourier analysis of the peak profiles have an intrinsic flaw, since it is necessary for each studied peak to be clearly isolated. If several peaks partially overlap, the resulting experimental signal corresponds to the sum of the elementary contributions, in which case it is impossible to extract the Fourier coefficients of each peak. This is why, in practice, the method suggested by Bertaut, and then by Warren and Averbach was essentially applied to crystals with a cubic... 

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