Fitting diffraction patterns

The structure refinement program for disordered carbons, which was recently developed by Shi et al [14,15] is ideally suited to studies of the powder diffraction patterns of graphitic carbons. By performing a least squares fit between the measured diffraction pattern and a theoretical calculation, parameters of the model structure are optimized. For graphitic carbon, the structure is well described by the two-layer model which was carefully described in section 2.1.3. 

Fig. 6. The X-ray diffraction patterns and calculated best fits from the structure refinement program for the samples MCMB2300, iVICMB2600 and iVfCMB2800.

The mechanism responsible for the formation of gas hydrates became clear when von Stackelberg and his school 42 49 in Bonn succeeded in determining the x-ray diffraction patterns of a number of gas hydrates and Claussen6 helped to formulate structural arrays fitting these patterns. Almost simultaneously Pauling and Marsh26 determined the crystal structure of chlorine hydrate. 

The Rietveld Fit of the Global Diffraction Pattern. The philosophy of the Rietveld method is to obtain the information relative to the crystalline phases by fitting the whole diffraction powder pattern with constraints imposed by crystallographic symmetry and cell composition. Differently from the non-structural least squared fitting methods, the Rietveld analysis uses the structural information and constraints to evaluate the diffraction pattern of the different phases constituting the diffraction experimental data. 

The poly(5-fnethyl-l, 4-hexadiene) fiber pattern (Figure 6) gave an identity period of 6.3 A, indicating a 3 isotactic helix structure. The X-ray diffraction pattern was not very sharp, which may be due to the difficulty of the side chain with a double bond to fit in a crystalline lattice. The crystallinity was determined to be 15% using the Hermans and Weidinger method (27). A Chloroform-soluble fraction free from catalyst residues showed no improvement in the sharpness of the X-ray diffraction pattern. These data show that the configuration of the 1,2-polymerization units in the homopolymer of 5-methyl-1,4-hexadiene is isotactic. 

Figure 5.8 A Debye-Scherrer powder camera for X-ray diffraction. The camera (a) consists of a long strip of photographic film fitted inside a disk. The sample (usually contained within a quartz capillary tube) is mounted vertically at the center of the camera and rotated slowly around its vertical axis. X-rays enter from the left, are scattered by the sample, and the undeflected part of the beam exits at the right. After about 24 hours the film is removed (b), and, following development, shows the diffraction pattern as a series of pairs of dark lines, symmetric about the exit slit. The diffraction angle (20) is measured from the film, and used to calculate the d spacings of the crystal from Bragg s law.

Fig. 50. Continuous random network fit to the X-ray diffraction pattern of H20(as) (from Ref. 82>). Experimental data and X Theory---...

Figure 24. X-ray diffraction pattern (in the inset the 2D image) of the polyethylene sample recovered by the laser-assisted high-pressure reaction in the pure liquid monomer. The two measured sharp lines nicely fit the polymer diffraction pattern having a orthorhombic cell (Pnam) defined by the lattice parameters reported in the figure.

The negative value of U in the fit in Fig. 7.11 signifies that the interaction between Li ions is attractive. Under an attractive interaction the ions can cluster, and the compound should separate into two phases at low temperatures. Fig. 7.12 shows one of the Bragg peaks in the X-ray diffraction pattern for Lio.sMoeScg as it cools (Dahn and McKinnon,... 

Electron crystallography of textured samples can benefit from the introduction of automatic or semi-automatic pattern indexing methods for the reconstruction of the three-dimensional reciprocal lattice from two-dimensional data and fitting procedures to model the observed diffraction pattern. Such automatic procedures had not been developed previously, but it is the purpose of this study to develop them now. All these features can contribute to extending the limits of traditional applications such as identification procedures, structure determination etc. 

The modeling of electron diffraction by the pattern decomposition method, for which no structural information is required, can be successfully applied for extraction of the diffraction information from the pattern. Several parameters can be refined during the procedure of decomposition, including the tilt angle of the specimen the unit cell parameters peak-shape parameters intensities. The procedure consists of fitting, usually with a least-squares refinement, a calculated model to the whole observed diffraction pattern. 

Figs. 1(a) and 1(b) show the best fits between calculated (solid line) and observed (dotted line) X-ray diffraction patterns for samples S3 and S4. 

Rietveld refinement [25, 26] is a method of whole pattern refinement, where a calculated diffraction pattern for a structure model is a least-squares fit to an observed diffraction pattern. Originally, it was used as a means of verifying proposed structure models. For zeolites, Rietveld refinement is still used for the same purpose and provides details of the structure including atomic positions of framework atoms and cation sittings. Data with accurate intensities and well-resolved peaks are needed for the most accurate work, and so often a synchrotron source is used for data collection since it can provide higher intensity and peak resolution than an in-house diffractometer. However, modern in-house diffractometers often provide good enough data for some refinements. 

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