Integral breadth

Probable errors in assigning the integral distribution curve, as indicated by scatter of the points in Fig. 57, are magnified in the process of taking the slope for the deduction of the differential distribution. Only the approximate location of the maximum and breadth of the latter are experimentally significant. 

In the simplest approach T is the full width of the peak (measured in radians) subtended by the half maximum intensity (FWHM) corrected for the instrumental broadening. The correction for instrumental broadening is very important and can be omitted only if the instrumental broadening is much less than the FWHM of the studied diffraction profile, which is always the case in presence of small nanoclusters. The integral breadth can be used in order to evaluate the crystallite size. In the case of Gaussian peak shape, it is ... 

The integral breadth of a ID, even and Fourier-transformable function h (r) is defined by... 

Then it follows from the slice theorem Eq. (2.38) for the integral breadth of the Fourier transformed function H (5)... 

In the field of scattering a simplified version of the Fourier breadth corollary Eq. (2.44) is known as the Scherrer equation21. As a result, the inverse of the integral breadth of a peak or reflection is the size of the crystal in the direction perpendicular to the netplanes that are related to the reflection. 

The cross-section of the primary X-ray beam is extended and not an ideal point. This fact results in a blurring of the recorded scattering pattern. By keeping the cross-section tiny, modern equipment is close to the point-focus collimation approximation - because, in general, the features of the scattering patterns are relatively broad. Care must be taken, if narrow peaks like equatorial streaks (cf. p. 166) are observed and discussed. The solution is either to desmear the scattering pattern or to correct the determined structure parameters for the integral breadth of the beam profile (Sect. 9.7). 

It is thus reasonable to make the slit height H of the detector slit wider than the integral breadth of the intrinsic primary beam profile. In this case the observed integral breadth equals H - and can be accurately determined from the measured primary beam profile. 

Line breadths are the fundamental quantities in this field of polymer analysis. As a consequence of the Fourier relation between structure and scattering these breadths are integral breadths, not full widths at half-maximum (FWHM). 

The effect of instrumental broadening can be eliminated by deconvolution (see p. 38) of the instrumental profile from the measured spectrum. If deconvolution shall be avoided one can make assumptions on the type19 of both the instrumental profile and of the remnant line profile. In this case the deconvolution can be carried out analytically, and the result is an algebraic relation between the integral breadths of instrumental and ideal peak profile. From such a relation a linearizing plot can be found (e.g., measured peak breadths vs. peak position ) in which the instrumental breadth effect can be eliminated (Sect. 8.2.5.8). 

Model Gaussian Peaks. If all the terms on the right-hand side of Eq. (8.13) can be modeled by Gaussians, the square of the integral breadth of the observed peak... 

On the other hand, lattice distortions of the second kind are considered. Assuming [127] that ID paracrystalline lattice distortions are described by a Gaussian normal distribution go (standard deviation ay, its Fourier transform Gd (.S ) = exp (—2n2ols2) describes the line broadening in reciprocal space. Utilizing the analytical mathematical relation for the scattering intensity of a ID paracrys-tal (cf. Sect. 8.7.3 and [127,128]), a relation for the integral breadth as a function of the peak position s can be derived [127,129]... 

An example for a linearization of integral peak breadths and the corresponding separation of size and distortion effects is sketched in Fig. 8.6. It is clear that at least... 

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...

Figure 8.44. Effect of paracrystalline distortions on a series of reflections in a scattering diagram after compensation of the decay according to POROD s law (lattice factor (1 /N) Z 2). The quadratic increase of integral breadths of the reflections is indicated by boxes of equal area and increasing integral breadth. L is the average long period...

As shown by Strobl [230], the integral breadths B in a series of reflections is increasing quadratically if (1) the structure evolution mechanism leads to a convolution polynomial, (2) the polydispersity remains moderate, (3) the rod-length distributions can be modeled by Gaussians (cf. Fig. 8.44). For the integral breadth it follows... 

In the standard setup W (y) is the profile of the primary beam in horizontal direction. In order to solve the smearing integral, the orientation distribution of the layer normals, g (), is approximated by a Poisson kernel121 and W (y) is approximated by a shape function with the integral breadth 2ymax of the primary beam perpendicular to the plane of incidence. In the simplified result... 

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... 

Let us consider a frequent problem the scattering of elongated voids or of mi-crofibrils is investigated. In such materials an equatorial streak is observed - similar to the one sketched in Fig. 9.6. If the voids were perfectly oriented, its integral breadth measured as a function of s 2... 

Separation of the two components is accomplished by means of data, in which the apparent azimuthal integral breadth... 

Here Bp describes the inevitable instrumental broadening by the known integral breadth of the primary beam16, and Bg is the true integral breadth of the orientation distribution. For the determination of (L) and Bg the relation is linearized... 

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... 

Problems arise, as the orientation distribution starts to split, but the split nature is not yet discernible. Thunemann [257] is discussing this problem in his thesis. He describes, how to determine the true tilt angle of the structural entities, and he determines the minimum tilt angle that is required for the split nature to become detectable (Fig. 9.8). We observe that, in practice, a split nature of Lorentzian orientation distributions (solid line) is detected earlier than a split nature of Gaussians - at least up to an apparent17 integral breadth of 70°. The reason is that Lorentzians are more pointed than Gaussians - in the vicinity of their maximum. 

Why is the determined integral breadth only an apparent one Lorentzians show a high background — and this background accumulates in particular, when Lorentzians are used as orientation distributions and thus are wrapped around the sphere. Thus die minimum of a Lorentzian orientation distribution on a sphere is not zero and die integral breadth that is determined by subtracting the minimum is only an apparent one. 

Figure 9.8. Minimum average tilt angle, structural entities measured with respect to the fiber axis at which the split nature of the orientation distribution becomes observable -plotted as a function of the integral breadth Bg of the orientation distribution g (Solid line g (tp) is a Lorentzian. Dashed Gaussian... 

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