Powder Pattern Calculation

Random structure methods have proved useful in solving structures from X-ray powder diffraction patterns. The unit cell can usually be found from these patterns, but the normal single-crystal techniques for solving the structure cannot be used. A variation on this technique, the reverse Monte Carlo method, includes in the cost function the difference between the observed powder diffraction pattern and the powder pattern calculated from the model (McGreevy 1997). It is, however, always necessary to include some chemical information if the correct structure is to be found. Various constraints can be added to the cost function, such as target coordination numbers or the deviation between the bond valence sum and atomic valence (Adams and Swenson 2000b Swenson and Adams 2001). 

This web site also contains an interactive powder pattern calculator that allows the user to change the input and variables for a powder pattern calculation. This book was typeset using UTj X with the standard computer modern typeface. The LT X file and associated postscript plots were generated with a C++ program written by M. M. J. Treacy. We wish to acknowledge the assistance and collaboration of the members of the IZA Structure Commission for proofreading the manuscript and providing additional information. We are indebted to our employers (Arizona State University, and Air Products Chemicals, Inc.) for support of this project, and to our wives Laura and Carol for their patience and support. 

Fig. 10 a Observed, calculated and difference X-ray powder diffraction profiles of Li3Ru04. b Theoretical X-ray powder pattern calculated using the structure shown in Fig. 9a [2]. Reproduced by permission of The Royal Society of Chemistry...

Figure 6. Effect of extra-framework material on XRD powder patterns. Calculated XRD patterns for ZSM-5 with and without ordered tetrapropylammonium and fluoride ions in the unit cell.

This web site also contains an interactive powder pattern calculator that allows the user to change the input and variables for a powder pattern calculation. 

Table 17.31 Available powder pattern calculation software (also refer to list of Rietveld structure refinement software).

Those peaks marked with ( ) are not present in the reported powder pattern of this compound but were confirmed as belonging to KFeS2 by comparison to a powder pattern calculated based on the published coordinates. ... 

Biochemical EPR samples are almost always collections of randomly oriented molecules (frozen) aqueous solutions in which each paramagnetic molecule points in a different direction. In order to generate simulations of these powder EPR spectra we have to calculate the individual spectrum for many different orientations and then add these all up to obtain the powder pattern. Numerical procedures that generate sufficient spectra to approximate a powder pattern are collectively known as walking the unit sphere algorithms. Here is the basic procedure ... 

The line shapes were calculated for the flipping motion with the two-site jump model described above, and the calculated spectra are shown in Fig. 11 for the higher temperature region. The experimental line shapes at 20 and 30° C are well reproduced showing the motional mode and rates obtained by T analysis are reasonable at least around these temperatures. Above 40°C the calculated line shapes are invariable and remain in the powder pattern undergoing a rapid flipping motion, while the experimental ones... 

Fig. 7 Solid state N-NMR spectra of GS-3/3 at 1 40 in oriented DMPC bilayers. The arrow indicates the new resonance emerging at high peptide concentration, from which the upright alignment of gramicidin S in the membrane was calculated. The dashed line corresponds to the respective powder pattern of the lyophilized peptide...

For the axially asymmetric powder pattern, the shielding anisotropy was calculated using the following expression... 

Powder patterns of crystals with axial symmetry yield the value of qQ but do not, of course, give the direction of the axis of symmetry. Line shapes to be expected for the magnetic resonance in this situation have been calculated (95) and for / = %, the shape is illustrated in Fig. 11 for polycrystalline corundum (a-AbOs). 

FIGURE 2. Calculated powder patterns for spin-1 (a) and spin-3/2 (b) nuclei with a dominating quadrupolar interaction (QUASAR simulation) tj is the asymmetry parameter of the quadmpolar interaction (equation (9))... 

In some systems for which there are known phases and even single crystal studies, the powder patterns for the known phases have not been deposited in the PDF. In this case there are at least three methods for obtaining the necessary patterns. The first is obvious the known phase is prepared and the powder diffraction pattern is recorded. However, it is not always possible to prepare single phase material of the known phase and so other methods must be employed. These other methods depend on the availability of crystal structure data for the desired phase. If the crystal structure is known, the complete powder pattern can be calculated. If only unit cell data are known, the rf-spacings of all possible lines for the phase can be determined from the relation-... 

The last method for producing standard patterns for phases not in the PDF is more involved. In many instances single crystals of unknown phases can be removed from reaction mixtures. If this is the case, a full three dimensional crystal structure analysis will yield the positions of all atoms in the structure. Once the crystal structure is known, it can be used to calculate the X-ray powder diffraction pattern for the phase. This powder diffraction information can then be used with confidence as a standard powder pattern. 

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