Pattern scale factor

Here 1)" is the observed and is the calculated intensity of a point i of the powder diffraction pattern, k is the pattern scale factor, which is usually set tk= because scattered intensity is measured on a relative scale and k is absorbed by the phase scale factor (e.g. see Eqs. 7.3 and 7.4, below), and n is the total number of the measured data points. Hence, a powder diffraction pattern in a digital format, in which scattered intensity at every... 

Since diffraction pattern has a scale factor LI all distances measured on the pattern should be recalculated into reciprocal. Following Zvyagin s notation one measures B k values on the texture pattern, which correspond to reciprocal values Hgoh by... 

Summarizing the results we can make a conclusion that all the methods of the centre calculation are almost equal in the precision. The maximum difference between the results obtained using described methods was 2 pixels, which is less than 0.003 A (the scale factor was about 750 to 900 pixels/A) for the given texture patterns. Such small difference has no significant influence on further calculations, which was proved by calculations performed on these texture patterns. 

Diffraction patterns may always be multiplied by a constant factor without changing the physics of the system. In MSLS this is a scaling factor which is needed to scale the observed and calculated intensities to the same order of magnitude. This scaling factor is one of the relinable parameters. However, one has to take care with the definition of this factor. It should be defined in such a way that it is not dependent on the change of any other parameter in the refinement procedure, in particularly the crystal thickness and the absorption parameter. Therefore the actual scaling factor, C, was expressed as function of c, the parameter which was used in the refinement process ... 

The second term leads to a vestigal powder pattern, scaled down by J2e. The rule of thumb becomes to reduce chemical shift anisotropy by a factor of 100, set the angle to within about 1/2 degree. 

This condition fixes the value of the scaling factor for the pattern Ce or Se, and the temporal behavior of this pattern is determined by exactly the same dispersion relation [equation (A3)] as for the circular case. Those modes which grow in time will form spatial patterns on the ellipse those which decay in time will not be seen. 

Due to these large deviations the direct application of the SQMF method to the B3 molecule may lead to incorrect assignment of the vibrational modes. Therefore we performed a preliminary SQMF calculation on the benzene molecule, where the normal modes are well ascribed [1] and discriminated by symmetry. The extracted scale factors correspond to the set of symmetry-adapted internal coordinates, as introduced by Wilson [1], The same scale factors were then applied to the B3 molecule, which was considered as a system constructed by three benzene molecules. The single C-N bonds in B3 were treated with the same scale factors as the single C-H bonds in the benzene molecule. The scale factors corresponding to the N-H bonds were initially set equal to 1. The as-obtained scaled force-field, after minor adjustment of the scaling factors, was employed to calculate the normal frequencies of B3. Fig. 2(b) shows the corresponding pattern of the calculated scaled frequencies. It can be seen that there... 

Toraya s Method. The WPPD as implemented by Toraya et al.11 decomposes the peak profiles and background functions to obtain the best fit to the experimental powder pattern of the individual pure phase data by least-squares refinement. The integrated intensities of the pure phases are then stored with the other refined parameters, such as the profile parameters and the unit-cell parameters of the phases to be quantified. During quantification step, the integrated intensity of the phase being quantified is scaled, as defined in Eq. (12.4), such that the total of the scale factors for the component phases sum to unity. The scale factors of the individual components are then refined by least-squares methods until a best fit is observed with respect to the pattern of unknown composition. 

Fig. 35. The polarizational (P), charge transfer (CT), and overall (P + CT) charge reorganization patterns for the four chemisorption arrangements of Figs. 33 and 34, obtained from the CSA calculations using the point charge approximation to dir. The arrow in the CT components indices the CSA predicted direction of the charge flow. In each row different scale factors are applied in the P, CT, and (P + CT) panels.

Such a refinement program was very useful in these cases but is in general of limited application, for it is only with very simple structures (both nuclear and magnetic) that a sufficient number of nuclear intensities can be accurately resolved at 4.2K to provide a basis for refinement and the determination of the scale factor. A more general refinement procedure has been recently introduced (69) which fits the measured profile of the powder diffraction pattern rather than individual intensities or structure factors. With data of high resolution obtained over a wide... 

The scattered intensity is nearly always measured in relative and not in absolute units, which necessarily introduces a proportionality coefficient, C. As we established before, when the phase angle is mr (n is an integer), the corresponding interference functions in Eq. 2.18 are reduced to U, U2 and Ui and they become zero otherwise. Hence, assuming that the volume of a crystalline material producing a diffraction pattern remains constant (this is always ensured in a properly arranged experiment), the proportionality coefficient C can be substituted by a scale factor K = CU U- Uz. 

Obviously, doing all of this is impractical, and in reality the comparison of the observed and calculated intensities is nearly always done after the former are normalized with respect to the latter using the so-called scale factor. As long as all observed intensities are measured under nearly identical conditions (which is relatively easy to achieve), the scale factor is a constant for each phase and is applicable to the entire diffraction pattern. 

Full profile refinement is computationally intense and employs the nonlinear least squares method (section 6.6), which requires a reasonable initial approximation of many fi ee variables. These usually include peak shape parameters, unit cell dimensions and coordinates of all atoms in the model of the crystal structure. Other unknowns (e.g. constant background, scale factor, overall atomic displacement parameter, etc.) may be simply guessed at the beginning and then effectively refined, as the least squares fit converges to a global minimum. When either Le Bail s or Pawley s techniques were employed to perform a full pattern decomposition prior to Rietveld refinement, it only makes sense to use suitably determined relevant parameters (background, peak shape, zero shift or sample displacement, and unit cell dimensions) as the initial approximation. 

Initial residuals (row one in Table 7.3), calculated using profile parameters determined from the full pattern decomposition employing Le Bail s algorithm and the default value of the scale factor (AT = 0.01), are quite... 

Figure 73. The observed and calculated diffraction patterns of LaNi4 g5Sno,i5. The scattered intensity was calculated using scale factor, instrumental and lattice parameters determined during Rietveld refinement, and guessed overall atomic displacement parameter fi = 0.5 A. All notations are identical to Figure 7.2.

In Eq. 7.10, /t is the number of different sets of powder diffraction data, is the number of data points collected in the 5" set, and is the scale factor for the x diffraction pattern, which appears because scattered intensity is measured on a relative scale. Other notations are identical to Eq. 7.3. Different scale factors, and K, are simple multipliers. Hence, they strongly correlate, and usually are not refined simultaneously. Constraining one of the scale factors (usually k, for the first diffraction data set) at 1 enables successful refinement of the phase scale K) and scale factors of all remaining sets of diffraction data ki, k, . .., k/,). Equations 7.4, 7.6 and 7.7 are modified in the same way as Eq. 7.3 for a combined Rietveld refinement. Furthermore, it is often the case that x-ray and neutron, or conventional x-ray and synchrotron data are used in combined refinements, therefore, the... 

Figure 7.9. The observed (Mo Ka radiation) and calculated powder diffraction patterns of LaNi4 85Sno.i5 after combined refinement of scale factors only. The inset shows the expanded view between 18.4 and 19.5° of 20 to illustrate the inaccuracy of lattice parameters.

Figure 7.19. The observed and calculated powder diffraction patterns of NiMn02(OH) after the initial Rietveld refinement with only the scale factor determined. The inset clarifies the...

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