Calculations pattern

Figure 30 shows a series of ealeulated patterns for carbon samples with a fraction, f, of carbon atoms in randomly oriented single layers, a fraction 2/3(1-f) in bilayers and a fraction l/3(l-f) in trilayers [12]. These cuiwes can be used to estimate the dependence of the ratio, R, defined by Fig. 29, on the single layer fraction. Figure 31 shows the dependence of R on single layer fraction for the calculated patterns in Fig. 30, and for another set of calculated patterns (not shown) where the fraction of carbon atoms in bilayers and trilayers was taken to be /2(l-f) [12]. Both curves in Fig. 31 clearly show that R decreases as the single layer content of the sample increases and is fairly insensitive to how the carbon is distributed in bilayers and trilayers. 

Fig. 31. The dependence of R on single-layer fraction for the calculated patterns of Fig. 30, and for a second set of calculations where the fraetion of earbon atoms m bilayers and trilayers is equal [12].

This practical question is of a familiar form. We wish to calculate the weight of product resulting from a specified weight of reactant. Our calculational pattern is applicable. 

Fig. 4.7 Powder X-ray diffraction patterns of two samples of Prl2. (a) Initial maximum temperature 1000°C. (b) 800°C. Calculated patterns I, V black lines, IV dashed, Pr2l5 grey.

Figures 2 (b) and (c) show a diffraction pattern obtained from a particle of diameter 1.5 nm and a diffraction pattern calculated for a multiply twinned, decahedral particle. The conclusion drawn from the study of many such observed and calculated patterns obtained from gold particles in the size range of 1.5 to 2 nm contained in a plastic film is that very few particles are multiply twinned, many have one or two twin planes but more than half are untwinned (16). This suggests that, at least for this type of specimen, there is no confirmation of the theoretical prediction that the multiply twinned form is the equilibrium state for very small particles.

Note For a rapid estimation of the isotopic patterns of chlorine and bromine the approximate isotope ratios Cl/ Cl = 3 1 and Br/ Br =1 1 yield good results. Visual comparison to calculated patterns is also well suited (Fig. 3.3). 

The complete powder XRD profile (either for an experimental pattern or a calculated pattern) is described in terms of the following components (1) the peak positions, (2) the background intensity distribution, (3) the peak widths, (4) the peak shapes, and (5) the peak intensities. The peak shape depends on characteristics of both the instrument and the sample, and different peak shape functions are appropriate under different circumstances. The most common peak shape for powder XRD is the pseudo-Voigt function, which represents a hybrid of Gaussian and Lorentzian character, although several other types of peak shape function may be applicable in different situations. These peak shape functions and the types of function commonly used to describe the 20-dependence of the peak width are described in detail elsewhere [22]. 

X-ray powder patterns determined in the present studies and from the literature are summarized in Table III. The observed patterns for Zeolon 100 Na and Zeolon 100 H (synthetic sodium and hydrogen large port mordenites) arc in excellent agreement with the calculated patterns for the Cmcm structure. 

Fig. 2.21. Calculated patterns of dipolar shifts ( ) [79] and —(5.) ( ) [81] values (proportional to the contact shifts) induced by lanthanide ions.

This fifth edition of the Zeolite Powder Pattern Collection contains calculated patterns of 226 zeolite materials representing 176 framework topologies - an increase of 43 new topologies since the fourth edition in 2001. 

The Collection is a source of reference patterns for pure crystalline phases. The data may be helpful in identifying known zeolitic materials and indexing their diffraction patterns. Because so many factors related to both the zeolite crystal and the diffraction instrument affect powder diffraction data, phase identification is not always straightforward and frequently requires additional data. Considerable care should be exercised in comparing calculated diffraction patterns to experimental patterns. For example, the use of fixed versus variable incident slits on a powder diffractometer can drastically change the relative intensities of a diffraction pattern, and it should be emphasized that calculated patterns are only as accurate as the structure refinements on which they are based. 

The data for all the materials presented in this work are sorted by increasing 26 value in the table. To identify an unknown material, measure the 26 values of the three most pronounced peaks (assigning strong weighting to any pronounced low-angle peaks, particularly those below about 10°) and find those materials with corresponding reflections at those values. This provides a starting point for a more detailed comparison of the experimental and calculated patterns. 

Because hydrated forms of natural zeolites or as-synthesized forms were used whenever refined atomic parameters were available, the plots should be easily comparable to experimental patterns. In some cases, only structure refinements of dehydrated or calcined forms were available. Significant differences in the intensities of low-angle peaks may be found when comparing the calculated pattern to experimental patterns of hydrated or as-synthesized materials. 

In the upper portion, the observed data are shown by the dots with error bars the calculated pattern is shown as the solid line pattern. In the central portion, the vertical markers show positions calculated for Bragg reflections. The lower portion is a plot of the y, difference, observed minus calculated. Gaussian profiles were used, 28 parameters were varied, and Rp = 12%, R p = 14%. 

The number of parameters adjusted was generally in the range 10-30 (7 to 51 in the x-ray cases reviewed by Young (6.)). The standard deviations in atom position parameters are typically 1-4 parts in 1,000 in the x-ray cases reviewed. Fig. 3 shows the x-ray patterns for a fluorapatite, CasfPOOsF, case in which 28 parameters were varied (20 of them structural), Rp= 12.1%, and RWn= 13.9% (4.). In this Fig., the observed pattern is indicated by the data points with vertical error bars, the calculated pattern is the continuous curve overlying them, the difference between observed and calculated patterns is shown by the lower curve, and the positions of Bragg peaks are shown by the vertical bars between the upper and lower curves. In this case the fit both looks to be very good and is so indicated by the R value. 

Figure 8.30 Rietveld refinement of the non-linear optically active inclusion compound 4-(4-dimethylamino-phenylazo)-l-methyl-pyridinium[MnCr(oxalate)3] 0.6MeCN. The experimental PXRD pattern is shown as crosses superimposed on a solid line representing the final calculated pattern. The difference between the two is the lower line, which should ideally be flat. The tick marks represent the positions of the individual Bragg peaks - the overlap particularly at high 29 is evident. In this case 7 = 1.45 % and 7 Bragg = 2.74 % representing good agreement (reprinted with permission from [37] 2001 American Chemical Society).

Chipera SJ, Bish DL. 2002. FULLPAT a full-pattern quantitative analysis program for X-ray powder diffraction using measured and calculated patterns. J. Appl. Cryst. 35 744-749. 

The aim of the method is to find the best agreement between measured v) and calculated y diffraction patterns. In the refinement procedure, the model is assumed optimum when the sum of the squares of the differences between the experimental and the calculated patterns, R, is a minimum according to a nonlinear optimization process [29], The sum of the squares of the differences between the experimental and the calculated patterns is given by... 

The calculated patterns were scaled by equating the intensities to the experimental values after integrating over the 11-50° 20... 

Filament-Winding. This requires a mandrel to shape the desired finished product. Continuous filament or woven tape is fed through a liquid resin bath to impregnate it, and then wound onto the mandrel in a calculated pattern to optimize the final properties (Table 15.27). The assembly is oven-cured. A collapsible mandrel can then be removed from the plastic product or the mandrel can be left as a part of the finished product. These are the strongest plastic products ever made. Typical products are pipes, tanks, and pressure bottles. Other suggested products... 

The Appendix gives calculated patterns based on the results of X-ray structure determinations for the clinker phases. For each pattern, a Reference Intensity Ratio (H23) is included. This is the integrated intensity of the strongest individual reflection (which may be a component of an overlap) relative to that of the strongest peak of corundum in a 1 1 mixture by... 

Once a tentative structure or model of the unit ceU contents is available, it is possible to calculate the diffraction pattern that would arise from such a structure. A comparison of the calculated pattern with the observed pattern is used to determine the deviation of the predicted structure from the actual structure. This information is used to further refine the deduced structure. This iterative refinement procedure is repeated... 

For example, the calculated powder patterns for four of the polymorphs of sul-phapyridine are given in Fig. 4.22. The calculated pattern represents that of a pure sample. Using some of the more sophisticated programs to calculate the powder pattern, one can assume the absence of preferred orientation or alternatively some specified degree of preferred orientation. The line shape can be varied to match experimentally observed line shapes. If the crystal structures of all the polymorphic forms (and impurities) in the mixture are known, then the diffraction patterns... 

Because of the general similarities in the diffraction patterns, and the lack of clearly resolvable distinguishing peaks, they employed the Rietveld method (Young 1993). In the Rietveld method, the entire experimental diffraction pattern for each solid phase is used as a basis for comparison. For structure determination using powder diffraction, this comparison is made with a structural model used to generate a calculated pattern. In quantitative analysis of polymorphic phases, the known crystal stmctures are used to generate the standard diffraction patterns and these are then refined against the experimental powder pattern of the mixture to obtain the relative amounts of the polymorphs. 

Fig. 7 (a) Observed Debye-Scherrer X-ray powder diffraction patterns (Cu-K ) for the large /, distance (ca. 8.0 A) material of composition ca. CjqAsF, (6) calculated pattern, (c) quartz capillary background... 

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