Microstrain broadening

The prerequisites for structure solution are to find the correct peak position and intensity. For the last step of SDPD, refinement of the structure (Rietveld refinement" ), it is also important to know the form of the diffraction peak, taking into account the instrumental contributions. Notably, in modern SDPD there is the ability to obtain information about deviation from the ideal structure. Crystallite size and microstrain broadening should be considered primarily as the contributors to the physical profile. All main Rietveld programs take into account these deviations from the ideal structure. 

Transmission electron micrographs (TEM) of submicrometer-size particles show faceted particles, and selected area electron diffraction (SAED) patterns of isolated particles show that they are formed by a small number of crystallites (Fig. 9.2.14a), This result is consistent with the mean size of the crystallites, which can be inferred from the x-ray diffraction lines broadening analysis using a William-son-Hall plot (35) in order to take into account the contribution of microstrains to the line broadening. Over the whole composition range, the mean crystallite size is in the range 40-60 nm for particles with a mean diameter in the range 200-300 nm (Table 9.2.5) (33). 

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 ... 

Attempts to consider a hkl dependence of the broadening in the Rietveld method were made by using different approaches involving ellipsoid, quartic and quadratic forms, either representing the modulation of the average sizes /,fc/and / /and microstrain hki, or used for a purely phenomenological qualitative fit without trying to extract any physical parameters. 

If the strain field is not homogeneous on the length scale of the crystallite size or smaller, according to Equation (9), different parts of the material diffract at slightly different angles, thus producing a broadened profile. Profile width and shape will evidently depend on the strain distribution across the sample. Considering the root mean square strain (or microstrain), Equation... 

As shown schematically in Figure 13.4c, the strain can change among different crystallites, e.g. as a consequence of plastic deformation in an elastically anisotropic medium, but can also vary across each crystallite, e.g. because of the presence of dislocations (the two terms are sometimes referred to as strain of the second and third kind, respectively). Notably, a strain broadening effect can be observed even if the macrostrain (mean value of the strain distribution) is zero (Figure 13.4c), as in a powder sample. Otherwise, the simultaneous effect of macrostrain and microstrain results in a shift and broadening of the diffraction profile (Figure 13.4d). 

This equation allows the spread in microstrain, AdId, to be calculated from the observed broadening assuming that size broadening is absent. 

Lundy and Eanes (1973) reported lengths in the c-axis direction of 200-400 A and microstrains of 3 to 4 x 10 determined from line broadening of the 002 and 004 apatite reflections from mineralized turkey leg tendon. They also tabulated earlier measurements of the dimensions of biological apatites in the c-axis direction. Wide-angle XRD also shows that older bone mineral is better crystallized than newly formed mineral. For example, Bonar et al. (1983) measured the improvement in crystallinity for bone from embryonic to 2-year-old chickens. They reported that the coherence length (neglecting microstrain) calculated from the Scherrer formula from the 002 peak increased from 107 to 199 A. In a similar study of bone from 3-week to 2-year-old rats (Ziv and Weiner 1994), the change in the same quantity was from approximately 200 to 280 A (means of 3 measurements). 

After calcination of HTT product at 350°C, a large broadening of XRD peaks of almost all components but Al° was observed, apparently caused by further oxidation of composites. For particles of calcined samples, "nucleation" of iron- enriched and Zr -enriched regions observed for the initial sample by TEM was detected as well (Fig.4b). This is accompanied by a large increase of microstrains density. So, apparently, calcination at a high temperature further oxidizes AC causing phase segregation (Fig.4c). Calcination at intermediate temperatures leads to more random distribution of Zr and Fe in AC. The activity of AC molded in alumina without any calcination is lower than that of a mildly calcined sample (Table 2). 

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. 

The reflection broadening in the XRD pattern is attributed to the contributions of the crystaUite size, microstrain and the instrument itself The crystal size D is inversely proportional to the peak breadth, according to the Scherrer s formula... 

Microstructural imperfections (lattice distortions, stacking faults) and the small size of crystallites (i.e. domains over which diffraction is coherent) are usually extracted from the integral breadth or a Fourier analysis of individual diffraction line profiles. Lattice distortion (microstrain) represents departure of atom position from an ideal structure. Crystallite sizes covered in line-broadening analysis are in the approximate range 20-1000 A. Stacking faults may occur in close-packed or layer structures, e.g. hexagonal Co and ZnO. The effect on line breadths is similar to that due to crystallite size, but there is usually a marked / fe/-dependence. Fourier coefficients for a reflection of order /, C( ,/), corrected from the instrumental contribution, are expressed as the product of real, order-independent, size coefficients A n) and complex, order-dependent, distortion coefficients C (n,l) [=A n,l)+iB n,l)]. Considering only the cosine coefficients A(n,l) [=A ( ).AD( ,/)] and a series expansion oiAP(n,l), A (n) and the microstrain e (n)) can be readily separated, if at least two orders of a reflection are available, e.g. from the equation... 

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). 

The refinement of oxide as well as oxyfluoride diffractograms has shown that line broadening is not due to the Gaussian contribution of microstrains (DST tan0 = 0). The crystallite sizes vary from 16 to 7 nm and decrease when Ca/Ce atomic ratio raises but are of the same order of magnitude for oxides and oxyfluorides for the same Ca/Ce atomic ratio. The estimated standard deviations corrected with the Berar factor are given in parentheses. 

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