Analysis of Diffraction Patterns

Despite the convenience, speed, and general precision offered by data analysis programs, it is important for the crystallographer to know what is being sought, and what consider- 

FIGURE 6.8 The lattice of asymmetric units, represented by the symbol for a question ( ), is the same in both (a) and (b). The corresponding reciprocal lattice is accordingly the same for both (c) and (d). If the unit cell is chosen as in (a) to be a primitive unit cell, then every lattice point in (c) is occupied. If, however, the real unit cell is chosen to be centered as in (h), then half of the reciprocal lattice points, indexed according to this centered cell, are systematically absent, and a checkerboard pattern of diffraction intensities is observed. 

We approach this problem systematically by asking a series of questions, and from their answers, either fixing certain properties or eliminating others as impossible. The questions we seek to answer, and the order of inquisition, is as follows  

What is the crystal class (triclinic, monoclinic, orthorhombic, trigonal, tetragonal, hexagonal or cubic)  

Is the crystal lattice primitive or centered That is, is it primitive P, C face centered, body centered I, or face centered F  

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Many investigations of small particles or of other materials may involve the collection and analysis of diffraction patterns from very large numbers of individual specimen regions. For small metal particles, for example, it may not be sufficient to obtain diffraction patterns from just a few particles unless there is reason to believe that all particles are of the same composition, structure, orientation and size or unless these parameters are not of interest. More commonly, it is of interest to obtain statistics on the variability of these parameters. The collection of such... 

For solid surfaces with crystalline structure, we can apply diffraction techniques for analysis. In a diffraction experiment the sample surface is irradiated with electrons, neutrons, atoms, or X-rays and the angular distribution of the outgoing intensity is detected. The analysis of diffraction patterns is a formidable task and in the first subsection we only introduce a simple case, which, nevertheless, contains the main features. A more general formalism for the interested reader is described in the Appendix. 

X-ray diffraction studies have shown that diffraction patterns of all studied copolymers possessing disilaindane fragments in the backbone display two diffraction maximums d = 7.24 - 8.81 A and d2 = 4.45 A, typical of amorphous polymers. Substitution of methyl framing by phenyl one in silaindane ring is accompanied by an increase of interchain distance. Analysis of diffraction patterns obtained proves the increase of interchain distance in copolymers with concentration of disilaindane fragments in them. 

Coming back to the detailed analysis of diffraction patterns, we note that such efforts can be in practice more complicated for real samples for different reasons. First of all, the crystallites (grains) inside a polycrystalline sample might have a preferred orientation (texture), and accordingly, the Bragg reflexes of all other orientations are extremely suppressed in their intensity compared to those expected from calculated structure factors. Such a behavior can be expected e.g. in the case of epitaxially grown thin films that adopt the structure or at least the orientation of the substrate. This is observed e.g. for the passive films on iron discussed above [4], and in part also for those on Ni [19] but also for electrodeposited metal films. 

WAXS, computer analysis of diffraction patterns and the ability of SAXS to distinguish the morphology of crystalline polymers. 

The analysis of XRPD patterns is an important tool studying the crystallographic structure and composition of powder compounds including the possibility to study deviation from ideal crystallinity, i.e. defects. Looking at an X-ray powder diffractogram the peak position reflects the crystallographic symmetry (unit cell size and shape) while the peak intensity is related to the unit cell composition (atomic positions). The shape of diffraction lines is related to defects , i.e. deviation from the ideal crystallinity finite crystallite size and strain lead to broadening of the XRPD lines so that the analysis of diffraction line shape may supply information about sample microstructure and defects distribution at the atomic level. 

These are two important special cases. The power and simplicity of diffraction pattern analysis (crystallography) for the analysis of regular structure is a result of Eq. (2.27) and Eq. (2.25). No information is lost if infinite abstract lattices are subjected to Fourier transformation. 

EDSA of thin polycrystalline films has several advantages First of all the availability of a wide beam (100-400 pm in diameter) which irradiates a large area with a large amount of micro-crystals of different orientations [1, 2]. This results into a special t5q)e of diffraction patterns (DP) (see Fig.l). Thus it is possible to extract from a single DP a full 3D data set of structure amplitudes. That allows one to perform a detailed structure analysis with good resolution for determining structure parameters, reconstruction of ESP and electron density. 

However, quite complicated algorithms are needed for extracting the data Ifom texture patterns. The analysis of texture patterns must be performed in a different way compared to regular electron diffraction patterns, due to different geometrical settings. Firstly, the centre of the... 

X-ray powder diffraction was recorded using a conventional x-ray powder diffractometer with Cu-Ka radiation. Polyimide film on which sample particles are deposited is glued on a glass sample holder with vacuum grease. Figure 1.6.9 shows the recorded diffraction pattern. An analysis of the pattern is made by comparing the lattice parameters and diffraction intensities of the particles and those of known iron compounds, and shows that the particles are Fe304. 

In summary, in the action of an optical microscope, " as was shown in Figure 6.8, diffracted beams result from a Fourier analysis of the pattern of light passing through the object. The image of the object obtained by recombining these diffracted beams is a Fourier synthesis of the Fourier analysis of the object. Thus a Fourier analysis is involved in diffraction and a Fourier synthesis is involved in the formation of an image. 

Consider the powder diffraction pattern shown in Figure 4.30 and answering Yes/No/Maybe Is this pattern suitable for phase identification Is the material suitable for crystal structure determination using powder diffraction Explain your reasoning. What other conclusions (if any) can be made from a visual analysis of this pattern ... 

The adjustable parameters for this model are refined by a least-squares minimization of the weighted differences between the observed and calculated intensities. The study by Loopstra and Rietveld on SrsUOg with 42 structural parameters clearly showed the power of this technique compared to the integrated intensity methods used earlier. This approach to the analysis of powder patterns has been so successful that it led to a renaissance in powder diffraction and this technique of treating powder diffraction data is now known as Rietveld refinement. ... 

Although, the powder method was developed as early as 1916 by Debye and Scherrer, for more than 50 years its use was almost exclusively limited to qualitative and semi-quantitative phase analysis and macroscopic stress measurements. The main reason for this can be found in what is known as the principal problem of powder diffraction accidental and systematic peak overlap caused by a projection of three-dimensional reciprocal space on to the one-dimensional 26 axis, leading to a strongly reduced information content compared to a single crystal data set. However, despite the loss of angular information, often sufficient information resides in the ID dataset to reconstruct the 3D structure. Indeed, quantitative analysis of the pattern using modern computers and software yields the wealth of additional information about the sample structure that is illustrated in Figure 1. Modern... 

Finally, a note on disorder of the membrane stacks and on attempts to correct for it in the analysis of diffraction data. Generally, two kinds of disorder are being discussed in crystal structure Disorder of the first kind refers to displacements of the structural elements (for example the one-dimensional unit cell of a membrane stack) from the ideal positions prescribed by the periodic lattice. The effect on the diffraction pattern is indistinguishable from that of thermal vibrations and may, therefore, be expressed as a Debye-Waller temperature factor so that the structure factor, expressed as a cosine series, includes a Gaussian terra, according to... 

Fig. 7. Dimers of water molecules localized by precise X ray powder diffraction analysis of a pattern recorded step by step (0.02° 2Q) for the hydrated form of SAPO-11 (from ref 37, fig. 2)...

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