Warren-Averbach method

We will give here a short overview of the most common XRPD techniques used to study the microstructure of materials, starting from the most used and simple Scherrer method to the quite complex Warren-Averbach method, which is able to extract all the information available on sample microstructure and defects. 

The same principle is the fundament of the WARREN-AVERBACH method (cf. Sect. 8.2.5.5) for the separation of size and distortion of structural entities. Thus die mathematics is partially identical. 

The Warren-Averbach method starts from this consideration and takes into account the simultaneous presence of size and strain effects. Once the IP has been properly considered e.g. by a deconvolution procedure), the Fourier expansion of the intrinsic profile can be written as ... 

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

Each of the single crystals in a polycrystalline sample is made up of a large number of mosaic blocks. These blocks are regions of the crystal which are so perfect that the dynamical theory of X-ray diffraction holds within them. The coherent domain size measured in the Warren-Averbach method is the average length of the mosaic blocks measured in a direction perpendicular to the diffracting planes. 

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. 

This property is readily established from the definition of Fourier transform and convolution. In scattering theory this theorem is the basis of methods for the separation of (particle) size from distortions (Stokes [27], Warren-Averbach [28,29] lattice distortion, Ruland [30-34] misorientation of anisotropic structural entities) of the scattering pattern. 

The indirect method described here returns the weight-average crystal size [121], irrespective of the model shape chosen. On the other hand, the direct Fourier inversion according to Warren-Averbach returns the number average of the crystal size distribution. 

Table 8.1. Integral breadth method according to WARREN-AVERBACH and the four basic possibilities for linearizing plots. All plots are tested for best linearization with the integral breadths from a set of peaks, and the best linearization is taken for structure parameter determination...

X-ray powder diffraction (XRD) was carried out using a Phillips MPD 1880 diffractometer, equipped with a Cu Ka source (A = 0.15418 nm), at 40 kV and 40 mA. The profiles were recorded at 0.02° (26) and step recording time of 5 s. The Ni(200) peak was fitted to a Pearson-VII profile shape function, as described elsewhere [5], with a residual error always lower than 1%. The method of Bertaut-Warren-Averbach (BWA) [11] was used to obtain CSDs and surface-average crystallite sizes, from which estimates of metal dispersion were obtained considering 0.065 nmVnickel atom and spherical crystallites [1]. Transmission electron microscopy (TEM) was carried out on a Jeol 200C working at 100 kV. The... 

X-ray diffraction (XRD) patterns were obtained by a Siemens D 500 powder diffractometer equipped with a graphite crystal monochromator using a Copper Ka X-ray radiation source. Experiments were run in step-scan mode witii a step interval of 0.02° 2e and a count rate of 1 second per step over the range 5° to 90° 20. The fraction of tetragonal to monoclinic form was determined by Rietveld method [11,12] while the crystallite size was determined by Warren-Averbach X-ray broadening method [13],... 

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. 

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... 

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]. 

From these relations, Warren and Averbach suggested, in the 1950s, a method to separate the effects of size from the effects of microstrains. They observed that the size term which is written in the form An = N /N3 does not depend on the diffraction order 1, whereas this order plays a role in the expression of the term... 

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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