Least squares diffraction

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. 

X-Ray diffraction from single crystals is the most direct and powerful experimental tool available to determine molecular structures and intermolecular interactions at atomic resolution. Monochromatic CuKa radiation of wavelength (X) 1.5418 A is commonly used to collect the X-ray intensities diffracted by the electrons in the crystal. The structure amplitudes, whose squares are the intensities of the reflections, coupled with their appropriate phases, are the basic ingredients to locate atomic positions. Because phases cannot be experimentally recorded, the phase problem has to be resolved by one of the well-known techniques the heavy-atom method, the direct method, anomalous dispersion, and isomorphous replacement.1 Once approximate phases of some strong reflections are obtained, the electron-density maps computed by Fourier summation, which requires both amplitudes and phases, lead to a partial solution of the crystal structure. Phases based on this initial structure can be used to include previously omitted reflections so that in a couple of trials, the entire structure is traced at a high resolution. Difference Fourier maps at this stage are helpful to locate ions and solvent molecules. Subsequent refinement of the crystal structure by well-known least-squares methods ensures reliable atomic coordinates and thermal parameters. 

So far no hypotheses are required concerning the true shape of the peak profile. Flowever, in order to avoid or reduce the difficulties related to the overlapping of the peaks, the experimental noise, the resolution of the data and the separation peak-background, the approach most frequently used fits by means of a least squared method the diffraction peaks using some suitable functions that allow the analytical Fourier transform, as, for example, Voigt or pseudo-Voigt functions (4) which are the more often used. 

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. 

We originally proposed NNM to be present in metallic beryllium [30] based on analysis of the X-ray diffraction data measured by Larsen and Hansen [24], Based on Fourier maps and elaborate multipole least-squares modeling, indisputable evidence... 

A preliminary least-squares refinement with the conventional, spherical-atom model indicated no disorder in the low-temperature structure, unlike what had been observed in a previous room-temperature study [4], which showed disorder in the butylic chain at Cl. The intensities were then analysed with various multipole models [12], using the VALRAY [13] set of programs, modified to allow the treatment of a structure as large as LR-B/081 the original maximum number of atoms and variables have been increased from 50 to 70 and from 349 to 1200, respectively. The final multipole model adopted to analyse the X-ray diffraction data is described here. 

Scolecite gave the opportunity to relate the electron density features of Si-O-Si and Si-O-AI bonds to the atomic environment and to the bonding geometry. After the multipolar density refinement against Ag Ka high resolution X-ray diffraction data, a kappa refinement was carried out to derive the atomic net charges in this compound. Several least-squares fit have been tested. The hat matrix method which is presented in this paper, has been particularly efficient in the estimation of reliable atomic net charges in scolecite. 

Figure 1 shows the powder X-ray diffraction (XRD) pattern of the as-prepared Li(Nio.4Coo.2Mno.4)02 material. All of the peaks could be indexed based on the a-NaFeC>2 structure (R 3 m). The lattice parameters in hexagonal setting obtained by the least square method were a=2.868A and c=14.25A. Since no second-phase diffraction peaks were observed from the surface-coated materials and it is unlikely that the A1 ions were incorporated into the lattice at the low heat-treatment temperature (300°C), it is considered that the particle surface was coated with amorphous aluminum oxide. 

For both structures, all final Si positions were obtained with reasonable accuracy (0.1 -0.2 A) by a 3D reconstruction of HRTEM images followed by a distance least-squares refinement. This kind of accuracy is sufficient for normal property analysis, such as catalysis, adsorption and separation, and as a starting point for structure refinement with X-ray powder diffraction data. The technique demonstrated here is general and can be applied not only to zeolites, but also to other complicated crystal structures. 

The success of the gas phase electron diffraction analysis of cis-and /ra 5-decalin (123) is another example of the use of MM calculations as an auxiliary technique. Minimum energy conformations and vibrational ampUtudes were calculated by both the Lifson and Boyd force fields (30,31) and were used as the starting values for refinement of the geometrical and vibrational parameters for the least-squares analysis. The results revealed no appreciable strain in cj5-decalin (123) other than that expected from gauche interactions. 

Butene exists as an equilibrium mixture of two conformations, Me-skew and Me-syn (21). The most reliable composition to date is 83 17% according to combined ED, microwave (MW), and ab initio MO analysis (133). This study includes the MM (CFF)-ED-MW analysis of this molecule for comparison, which gave a final skew/syn ratio of 80 20. The molecular orbital constrained electron diffraction (MOCED) results appear to agree better with the observed data than does the MM constrained analysis, the R value of the least-squares analysis of the latter being 20% higher than that of MOCED. However, one may ask whether such a small difference in R values justifies the enormous difference in computer time between the ab initio (about 200 hr on an IBM 370/155) and MM (less than a minute) methods used in this work. 

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. 

Most of the unknown structures is determined from single crystal diffraction and refined from powder diffraction. Refinement is done with the Rietveld method, which is a least square fitting of the computed pattern to the measured one, while structure parameters are treated as the primary fitting parameters. This is in contrast to the procedure in pattern decomposition, which is outlined above (where not the structure parameters, but the peak intensities were the primary fitting parameters). Beside the... 

Figure 15. Crystal structure of a-Tl2Se solved in projeetion via direct methods using quantified intensities from the selected area electron diffraction pattern shown in (a) [film data]. The potential map (E-map) in (b) was used to eonstruet an initial structural model which was later improved by kinematical least-squares (LS) refinement (c). Note that the potential of the selenium atoms in (c) appear after LS-refinement somewhat stronger than the surrounding titanium atoms (see the structural model in figure lOd). The average effective thiekness of the investigated thiekness of the crystal is about 230 A [22].

By far the best way to refine structures using electron diffraction data is to use multislice calculations. These will be discussed in the next chapter. However, some useful information can be obtained by regular crystallographic least squares with the assumption of kinematic data. 

Structure determination from X-ray and neutron diffraction data is a standard procedure. Starting with a rough model, the accurate structure is determined using a least-squares structure refinement, which is based on kinematic diffraction and in which the differences between calculated and experimental intensities are minimized. X-ray and neutron diffraction are not applicable to all crystals. To determine crystal structures of thin layers on a substrate or small precipitates in a matrix (see figure 1) only electron diffraction (ED) can lead you to the crystal structure. 

We have developed a software package MSLS [1], in which multi-slice calculation software is combined with least squares refinement software used in X-ray crystallography. With multi-slice calculations which are standardly used for image calculations of HREM images, dynamic diffraction is taken into account explicitly. 

J. Jansen, D. Tang, H.W. Zandbergen and H. Schenk, MSLS, a least-squares procedure for accurate crystal structure refinement from dynamical electron diffraction patterns. Acta Cryst. A54 91-101,1998. 

Therefore, uniaxially oriented samples should be prepared for this purpose, which give so-called fiber pattern in X-ray diffraction. The diffraction intensities from the PPX specimen of P-form, which had been elongated 6 times at 285°C, were measured by an ordinary photographic method. The reflections were indexed on the basis of the lattice constants a=ft=2.052nm, c(chain axis)=0.655nm, a=P=90°, and y=120°. Inseparable reflections were used in the lump in the computation by the least square method. 

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