Digital patterns

Figure 23. The 2D illustration of the digital pattern analysis for computing the Euler characteristic. The local curvature variables xe —1,0, +1 are assigned to the each lattice site at the boundaries of black pixels. The Euler characteristic is a sum of local curvature variables...

The formulas (120) and (127) define a set of the local curvature variables that can be used for the digital pattern analysis in the case of an arbitrary lattice. Calculation of the curvature distribution is, in principle, impossible within the digital pattern methods. 

Figure 38. An example when the Euler characteristic cannot be correctly determined by using the digital pattern analysis, because the boundaries of the domains after the thresholding procedure cannot be specified.

In conclusion, the digital pattern method is very efficient computationally but fails to calculate the Euler characteristic for the cases similar to the one shown in Fig. 38. Such cases are typical, however, when the data are noisy or when the field values are distributed around the threshold value. 

The disagreement of those two methods can be used to characterize quantitatively the double-separated morphology. One should simply calculate the Euler characteristic once by using the Euler formula (130), and then calculate it a second time by using the digital pattern method. The difference between obtained quantities will give the number of double-separated structures. 

Recent developments and prospects of these methods have been discussed in a chapter by Schneider et al. (2001). It was underlined that these methods are widely applied for the characterization of crystalline materials (phase identification, quantitative analysis, determination of structure imperfections, crystal structure determination and analysis of 3D microstructural properties). Phase identification was traditionally based on a comparison of observed data with interplanar spacings and relative intensities (d and T) listed for crystalline materials. More recent search-match procedures, based on digitized patterns, and Powder Diffraction File (International Centre for Diffraction Data, USA.) containing powder data for hundreds of thousands substances may result in a fast efficient qualitative analysis. The determination of the amounts of different phases present in a multi-component sample (quantitative analysis) is based on the so-called Rietveld method. Procedures for pattern indexing, structure solution and refinement of structure model are based on the same method. 

Figure 3. Whole mount of a chicken wing showing the skeletal pattern following implantation of a bead soaked in retinoic acid to the anterior margin. The digit pattern is duplicated 4 3 2 2 3 4.

In summary, inkjet offers non-contact digital patterning of conductive features in an additive process with the possibUity of utilizing flexible and temperature sensitive substrates. The associated process involves low cost set-up and maintenance, provides relatively high resolution, and allows for quick turnover and customization. Overall, IJ printing wiU ultimately enable production of a broad range of electronic components that are made completely by printing processes. 

These relationships are not affected by the tilting angle of the tube (see below). Because d2 and d correspond to the diffraction fines fiaving relatively strong intensities and are further from the equatorial fine, they are used in our study instead of di to reduce the error. The distances can be measured precisely from the digitized patterns. The errors are estimated to be <1% for the diameter determination and <0.2° for the chiral angle. 

Figure 4.8. Automatic peak search conducted using a second derivative method (top) and manually corrected reduced pattern (bottom). The upward arrow placed on the digitized pattern shows a false peak (which was eliminated manually) and the downward arrows show the missed peaks (which were added manually).

A digitized representation of powder data is quite compact and is especially convenient for comparison with other patterns, provided a suitable database is available. In addition to a digitized pattern, each entry in such a database may (and usually does) contain symmetry, imit cell dimensions, and other useful information phase name, chemical composition, references, basic physical and chemical properties, etc. Powder diffraction databases find substantial use in both simple identification of compounds (qualitative analysis) and in quantitative determination of the amounts of crystalline phases present in a mixture (quantitative analysis). 

The diffraction pattern from a single crystal is also unique but due to the complexity of a three-dimensional distribution of intensities, phase recognition is difficult to formalize. Powder data are one-dimensional, and they can be converted into digitized patterns, which are in a way, unique barcodes enabling automated pattern recognition. 

Phase identification using powder diffraction data requires a comparison of several key features present in its digitized pattern with known compounds/phases. This is usually achieved by searching powder diffraction database(s) for records, which match experimentally measured and digitized pattern. Thus, a powder diffraction database or at least its subset should be available in addition to a suitable search-and-match algorithm. 

Digitized pattern. Each observed Bragg reflection is listed with ... 

When experimental data remain unidentified using a digitized pattern-based search-match, different databases should be checked before drawing a conclusion that a material is new. Continuing searches, however, usually require unit cell dimensions and therefore, a powder pattern should be indexed prior to the search. There are a variety of databases dedicated to different classes of compounds and containing different information as shown in Table 4.3. ... 

Figure 4.22. Indexed PDF card (also see Figure 4.20, middle). Every observed Bragg reflection has been indexed and the corresponding F30 figure of merit (see section 5.5.1 in Chapter 5) is excellent. Based on these and other established criteria, the quality mark assigned by the ICDD editor is Quality Data , which usually is a good indicator that the included digitized pattern may be trusted in positive phase identification.

Figure 4.23. The results of a qualitative analysis of a multiple phase sample. Three crystalline phases are clearly identifiable lithium silicate - Li2Si03, silicon oxide - SiOj (quartz), and a different pol)imorph of silicon oxide - tridymite. A low quality diffraction pattern collected during a fast experiment was employed in this example. The data shown on top were smoothed, the background was subtracted, and the Ktt2 components were stripped before the digitized pattern (shown below the smoothed profile) was obtained using an automatic peak search. Note, that many weak Bragg reflections were missed in the peak search,...

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