Warren-Averbach Fourier

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. 

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. 

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 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 ... 

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... 

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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