Microstrain function

The first term is related to the (surface weighted) average size, whereas the second is a function of the microstrains. The first term is the same for all peaks, whereas the second depends on the interplanar spacing, "d", of the diffraction planes, and is proportional to 1/d. The... 

As in the bar screen, the value of the coefficient can be easily determined experimentally from an existing microstrainer. In the absence of experimentally determined data, a value of 0.60 may be assumed for Q. Also, from the equation, as the microstrainer clogs, the value of A2 will progressively decrease thus the head loss rises to infinity, whereupon, the strainer ceases to function. Although the previous equation has been derived for microstrainers, it equally applies to ordinary screens where the approach velocity is negligible. 

It is worth noting that unlike the instrumental and wavelength dispersion functions, the broadening effects introduced by the physical state of the specimen may be of interest in materials characterization. Thus, effects of the average crystallite size (x) and microstrain (s) on Bragg peak broadening (P, in radians) can be described in the first approximation as follows ... 

Figure 13.4 PD peak profile for zero strain (no macrostrain and no microstrain) (a), (tensile) macrostrain (b), microstrain (c) and combined effect of micro strain and macrostrain (d). Strain (s) is plotted on the left as a function of the position within a material microstructure sketched in the middle drawing.

The WA plot is shown in Figure 13.8 for the stabilized-zirconia powder of Figure 13.7 In(Ai) is plotted as a function of for three reflections of the (0//) family, (Oil), (022) and (033). According to Equation (18), from the intercept and slope of the regression lines, respectively, size Fourier coefficients and microstrain can be obtained for different Fourier lengths, L. 

This relation and equation [5.107] serve as the basis for the use of Fourier series analysis of diffraction peak profiles to quantitatively characterize stmctural defects found in crystals. We will see in the following chapter how this result can help us determine, in particular, the size and microstrain distribution functions. 

An alternative approach consists of working directly with the measured profile h(x), by expressing it as a Fourier series that includes the various components associated with each of the effects that modify the experimental profile [SCA 02] (instrumental function, size effect, microstrains, etc.). The intensity distribution h(x) or 1(20) is then expressed from equation [5.98] which was obtained in Chapter 5 by including in this equation the Fourier transform G(x) of the instmmental function, and finally ... 

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 ... 

Regardless of whether this general description or the one described earlier is used, the study of size and microstrains will always reqnire determining the instrument s own contribution. As we showed in the equations above, this contribution is expressed through the values of the parameters U, V, W and X. Therefore, in this case, the instrumental function will be determined by refining these parameters based on a pattern obtained with a standard sample that does not cause any increase in peak width. 

This means that Voigt function fitting makes it possible to determine the size and the microstrain rate of each family of planes, along the direction perpendicular to this family of planes. 

Figure 6.16. Size and microstrain distributions measured by Fourier analysis ofpeaks fitted with Voigt functions...

Over the past few years, an alternative method has appeared. This time, it consists of directly comparing the measured signal with the one that was calculated by the successive convolutions of functions expressing the different effects of the instmment and the microstracture. We already mentioned this modeling approach in the first part of this chapter. We will now explain how it is implemented in order to determine the size and the microstrains. 

More recently, measurements have been developed based on the complete evaluation of the intensity distribution over the entire diffraction spot. This reciprocal space mapping [FEW 97] can be used to give an account of all the microstractural effects that have an influence on the diffraction signal. Thus, it is possible to obtain the distribution functions of orientation, size, morphology and the size distribution of the diffracting domains, as well as the microstrains. 

Fig. 23.15 Lattice parameter and microstrain measurements as a function of the heat-treatment temperature. Powders of stannic oxide were prepared from precipitated hydrous 3-stannic acid by oxidizing tin granules with nitric acid, evaporating the liquid and firing the powders at different temperatures under oxygen atmosphere (Data from Leite et al. 2002)...

The Thompson-Cox-Hastings function is often used to refine profiles with broad diffraction peaks because it is the more appropriate model for line-broadening analysis where the Lorentzian and Gaussian contributions for crystallite size and for microstrains are weighted. So in this case, the peak shape is simulated by the pseudo-Voigt function, which is a Unear combination of a Gaussian and a Lorentzian function (Table 8.5). 

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