Strain broadening

Most real crystals contain imperfections producing local distortions of the lattice, resulting in a non-homogeneous strain field. The effect on position, shape and extension of reciprocal space points, and consequently on PD peak profiles, is usually more complex than that of the domain size. A formal treatment of the strain broadening is beyond the scope of the present book interested readers can refer to the cited literature. In the following a simplified, heuristic approach is proposed. 

First consider the effect of a macroscopically homogeneous strain (or macrostrain), expressed as z = ts.djd. By differentiating Bragg s law (assuming a constant wavelength)  

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 

Microstrain has a rather peculiar effect on reciprocal lattice points - taking into account the cos 6/1 transformation factor [Equation (5)] from 26 to reciprocal space. Equation (10) implies that  

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There is another difficulty related to the application of these anal5(tical techniques, especially dealing with very small particles it is often impossible to find a couple of peaks belonging to the same family, so it is impossible to separate the size and strain broadenings. In these cases, the broadening is usually assumed to be only due to the size effect, but it is important to remember that we are probably underestimating the crystallite dimensions. 

It is worth to be noted that these definitions of first- and second-order distortions according to Warren-Averbach are model-free. From a linear or a quadratic increase of peak breadths it can neither be concluded in reverse that strain broadening, nor that paracrystalline disorder were detected. 

Distortions of the First Kind and Strain Broadening. Rolled and drawn metals exhibit a broadening of peaks that increases linearly with the peak order. The first explanation of this observation and its theoretical treatment goes back to Kochendorfer [112-115], In the field of SAXS similar considerations have first... 

Resorting to reasoning of Kochendorfer ([112] and [115], p. 463) an approximation for the breadth of the lattice distortion term due to small amounts of strain broadening... 

Hagen, W.R. 1981. Dislocation strain broadening as a source of anisotropic linewidth and asymmetrical lineshape in the electron paramagnetic resonance spectrum of metal-loproteins and related systems. Journal of Magnetic Resonance 44 447-469. 

For pulses shorter than v lps the sepctrum will be broader and less pronounced that for pulses longer than 1 ps. Otherwise stated, the electron states immediately after excitation find themselves in a "foreign lattice" and this introduces strains which take finite time to diffuse before the ir-electrons accomodate themselves in a new lattice. These initial strains broaden the spectrum in the initial stage of excitation (up to 1 ps) for time longer than 1 ps the spectrum coalesces to the one observed with very long pul-esor stationnary sources. 

Rietveld minimization continued by including porosity effects, which were refined with the majority of other parameters fixed and then released. The fit improves and Rwp is reduced to 15 %. To this point, the refinement of the grain size and strain broadening effects and their anisotropy, as well as the coordinates of atoms in the organic molecule could not be easily conducted due to noticeable correlations the resulting shifts of fi ee least squares parameters were forcing the solution out of a global minimum. 

Plus strain broadening, Y, then asymmetry and sample displacement 7.3 9.7 6.2 22.7... 

A second difficulty is the anisotropic size and strain broadening of the diffraction peaks, which is considerable as well, but highly correlated with the broadening due to stacking disorder. A de-convolution is not possible without determining the anisotropic size distribution, e.g., with electron microscopy, which is difficult to perform due to technical temperature and pressure constraints making it impossible to observe a stable sample of ice Ic. 

Therefore, while domain size gives the same effect i.e. same width and shape) for all reciprocal space points (Figure 13.2), strain broadening varies from point to point, generally increasing with the diffraction order (Figure 13.5). 

Figure 13.5 Strain broadening effect in reciprocal space. In this schematic picture, according to Equation (10), points spread increasingly with the distance from the origin.

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). 

BAL 93] BALZAR D., LEDBETTER H., Voigt function modeling in Fourier analysis of size and strain broadened x-ray diffraction peaks , J. Appl. Cryst, vol. 26, p. 97-... 

UNG 99] UNGAR T., The dislocation based model of strain broadening in x-ray line profile analysis , in SNYDER R.L., FIALA J., BUNGE H.J. (eds.). Defect and microstructure analysis by diffraction, lUCr Monographs on crystallography, no. 10, Oxford University Press, p. 264-317,1999. 

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