Integral breadth method

3 TRADITIONAL VERSUS INNOVATIVE METHODS 13.3.1 Integral Breadth Methods 

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  

According to this expression, the slope of a regression line in a plot of fi s) versus d j i (known as a WH plot) gives 2e, whereas the intercept gives the inverse of the apparent size, L v- It is implicitly assumed that P(s) refers to the intrinsic profile, i.e. the IP component has been removed. 

Equation (14) can also be modified to include terms accounting for fault-ing. It is also possible to consider the strain anisotropy arising from the 

The weak point of Equation (14) is that the additivity of integral breadths is an arbitrary choice, valid only under rather restrictive conditions. In fact, owing to the mechanism of convolution of various profile components shown by Equation (13), the way the IB components add up depends on specific features of the broadening sources (and relevant profile components). In particular. Equation (14) can be justified if I s) and P s) both have a Lorentzian shape. For Gaussian profiles, instead, one should write  

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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...

Buchanan, McCullough, and Miller (22) found that the non-transform integral breadth methods effectively give a weight average rather than a number average Li. The weight... 

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]. 

Aside from a few authors [BER 93, GRO 98b, SAN 97] who still implement it, the variance method is hardly used anymore. This chapter will deal with the integral breadth method and the Fourier analysis. 

Microstructural study using the integral breadth method... 

The mean sizes and standard deviations characterizing the distributions shown in Figure 6.17 are listed in Table 6.2. These values ate compared with those given before in Table 6.1, which were obtained by using the integral breadth method. 

As you can see, the mean sizes obtained by modeling are slightly greater than those found from the integral breadth method. Naturally, the main advantage of the modeling method is that the distribution functions of the microstractural characteristics are known. 

History. Wilke [129] considers the case that different orders of a reflection are observed and that the orientation distribution can be analytically described by a Gaussian on the orientation sphere. He shows how the apparent increase of the integral breadth with the order of the reflection can be used to separate misorientation effects from size effects. Ruland [30-34] generalizes this concept. He considers various analytical orientation distribution functions [9,84,124] and deduces that the method can be used if only a single reflection is sufficiently extended in radial direction, as is frequently the case with the streak-shaped reflections of the anisotropic... 

Figure 9.7. Separation of misorientation (Bg) and extension of the structural entities (1/ (L)) for known breadth of the primary beam (Bp) according to Ruland s streak method. The perfect linearization of the observed azimuthal integral breadth measured as a function of arc radius, s, shows that the orientation distribution is approximated by a Lorentzian with an azimuthal breadth Bs... 

Non-transform methods. The non-transfona methods make use of Jones type relations thus, if 3s aHd 3d are the integral breadths due to size and distortion respectively, then, the observed profile 3 is given by... 

On the basis of the paracrystalllne theory, the Cauchy-plot method can be employed in line-broadening analysis (12). In this method, the integral breadth of the first order reflection is approximated by... 

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. 

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. 

According to the integral breadth (IB) method based on the paraoystalline theray the IB of the 001 reflection (AS ) of the Cauthy type b given by ... 

This step allows a variety of methodologies to be used to assess the safety systems identified in Step 4. The chosen method(s) and the breadth and detail use will depend on the safety Integrity criteria adopted and are likely to be the ones listed for Step 2, but conducted at a more detailed level. 

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