A Analysis of diffraction patterns

Here we introduce the more general theory of diffraction at crystal surfaces. First, we analyze for which directions of the outgoing radiation we get constructive interference and observe diffraction peaks . In the second part we discuss what determines the intensities of these maxima. 

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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. 

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. 

A serious problem, however, for both paramagnetic and coherent magnetic scattering measurments, at least until higher reflectivity polarizers are available or alternative techniques of polarization analysis are developed, is the low intensity obtained after reflection from the sample as well as from two polarizing crystals. Even for location of apparatus at a high flux reactor, with current techniques the measurement of covalent spin reductions in powder samples can be made as efficiently by profile analysis of a conventional powder diffraction pattern as by polarization analysis. Polarized beam methods including polarization analysis will however, be essential for the determination of form factors. 

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. 

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,...

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 ... 

A white crystalline powder, prepared by hydrothermal treatment at 200°C of a mixture of molybdic acid, H2M0O4, and methylammonium ma) chloride, CH3NH3CI, taken in a 1 2 molar ratio and acidified with hydrochloric acid, HCl, to pH = 3.5, resulted in a complex powder diffraction pattern shown in Figure 6.29. It was indexed in the monoclinic crystal system as was discussed in section 5.12.2. The space group C2/c (or its acentric subgroup Cc) was established from the analysis of the systematic absences, and the unit cell dimensions were refined using 120 resolved reflections below 20 = 60° a = 23.0648(6) A, b = 5.5134(2) A, c = 19.5609(5) A, p = 122.931(1)°, and the sample displacement 8 = -0.098(3) mm for a 250 mm goniometer radius. The unit cell volume is 2087.8 A. ... 

Early data analysis attempted to extract values of the individual structure factors from peak envelopes and then apply standard single crystal methods to obtain structural information. This approach was severely limited because the relatively broad peaks in a powder pattern resulted in substantial reflection overlap and the number of usable structure factors that could be obtained in this way was very small. Consequently, only very simple crystal structures could be examined by this method. For example, the neutron diffraction study of defects in CaF2-YF3 fluorite solid solutions used 20 reflection intensities to determine values for eight structural parameters. To overcome this limitation, H. M. Rietveld realized that a neutron powder diffraction pattern is a smooth curve that consists of Gaussian peaks on top of a smooth background... 

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