The Williamson-Hall method

In 1953, Williamson and Hall [WIL 53] snggested a simple method for solving this problem. It works by considering that both the limited size of the ciystals and the presence of crystallographic distortions lead to Lorentzian intensity distributions. 

This is a very simplistic assnmption, of conrse, but it provides a simple way of dealing with the problem. As we have already said, the convolntion prodnct of two Lorentzian functions is a Lorentzian function and, additionally, the residnal breadth is the sum of the elementary breadths caused by each of these effects. If we denote by Pp the pure breadth and by P and P° the breadths related to size and microstrains, respectively, then we obtain  

If we plot (P cos 0)/X according to (sin 0)/X, we get a straight line with a y-intercept equal to the inverse of the size and a slope equal to the value of the microstrains. This plot is known as the Williamson-Hall plot. 

The y-intercept enables us to determine a mean value of the size, which is in this case about 22 nm. The slope of this line is small, which shows that the sample contains a small amount of microstrains. The measured value is 0.13%. 

This approximated method makes it possible to obtain a qualitative mean value 

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As a natural extension of the methods shown in Section 13.2, IB expressions for the various sources of line broadening can be combined. This is the basis of the Williamson-Hall method, introduced in the late 1940s.Considering Equations (7) and (10), size and strain contributions can be combined as ... 

An empirical method that is not related to a rigorous treatment of the convolution of a diffraction profile by size and strain is the Williamson-Hall analysis. This method is suitable for substances characterized by a large number of diffraction peaks and for highly defective samples for which analytical procedures bring upon problems with background definition. The method involves plotting of reciprocal breadth ((3 ) (FWHM) in units of the 20 scale versus the reciprocal positions (d ) of all peaks of a phase. The intercept yields the particle size and the slope the "apparent strain" 2r. The required quantities are defined as follows ... 

It was well known that Williamson-Hall method is more accurate method to calculate crystallite size as compared to Debye-Scherrer method. The Fig. 2 represents the Williamson-Hall (W-H) plot for YP04 Eu nanostructure phosphor. As shown in Fig. 2, the Y- intercept is 0.0027 taking X as 0.154 nm, the grain size was found to be around 62 nm. The calculated particle size was in good concurrence with the Debye-Scherrer data. The small variation in the size of grains calculated by Debye-Scherrer and W-H method was due to the fact that in Debye-Scherrer formula strain component was assumed to be zero and the diffraction peak broadening was assumed to be due to reduced grain size only. 

In Equation 4.1, the factor fiF(0) is included, which is the peak profile function, that describes particle size broadening and other sources of peak broadening. The XRD method can be used as well for the measurement of the crystallite size of powders by applying the Scherrer-Williamson-Hall methodology [4,35], In this methodology, the FWHM of a diffraction peak, p, is affected by two types of defects, that is, the dislocations, which are related to the stress of the sample, and the grain size. It is possible to write [35]... 

The crystallite size of the BaCe0 95Yb0 05O3 5 perovskite powder was calculated following the method previously explained, that is, the Scherrer-Williamson-Hall methodology [4,35,36], The calculated radius of the perovskite crystallite, considered as spherical particles, was a = = 71 + I nm. 

Figure 6.4. Williamson-Hall plots obtained by using the Langford method based on diffraction peak fitting of cerium oxide...

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