Pawley method

After the unit cell has been determined, the next stage is to prepare the intensity data for space group determination and structure solution using an appropriate profile fitting technique such as the Pawley method [30] or the Le Bail method [31]. The aim of this stage of the structure determination process is to fit the complete experimental powder XRD profile by refinement of variables that describe ... 

There are three common methods used for full-profile fitting, namely, Le Bail method,Pawley method, and... 

The Pawley method is a method of least squares analysis where parameters that depend on the intensities, namely, atomic coordinates, displacement parameters, etc., are ignored and refinement is against the intensities themselves. 

In Direct methods, the intensities are extracted from an indexed powder diffraction pattern by profile fitting procedures such as the Le Bail method and Pawley method. Then the integrated intensities obtained are corrected for Torentz polarization and normalized. These corrected intensity values are then subjected to routine Direct-method procedures. The structural model obtained is completed using difference Fourier maps. This method works successfully when a sufficient number of intensities could be extracted from the powder diffraction pattern. 

Pawley method, as well as the Rietveld method. Also, it is noteworthy that even very fast scan data allow a precise analysis because of the high signal-to-noise ratio possible with the help of a high-speed detector (Figure 5). 

The pressure dependence of wavenumbers has been investigated theoretically by LD methods on the basis of a Buckingham 6-exp potential. In the studies of Pawley and Mika [140] and Dows [111] the molecules were treated as rigid bodies in order to obtain the external modes as a function of pressure. Kurittu also studied the external and internal modes [141] using his deformable molecule model [116]. The force constants of the intramolecular potential (modified UBFF) were obtained by fitting to the experimental wavenumbers. The results of these studies are in qualitative agreement with the experimental findings. 

Experimental tilt angles have usually an accuracy of at best 3°, leading to an error of about 0.1 A in cell axes. The calculated third cell axis will show a higher deviation. If possible an internal standard should be used for calibration purposes but a higher accuracy will be obtained with a Pawley fit (e.g. fit forP CuPc in Fig. 6) from x-ray powder diffraction data [11]. Especially for packing energy minimization used for simulation methods it is essential to determine the cell parameters as precise as possible. In the case of polymorphism, it is essential to use x-ray powder diffraction to ensure that bulk and investigated nano crystals represent the same modifications. 

Pawley, G. S., in Advances in Structure Research by Diffraction Methods," Vol. 4, Pergamon, Vieweg, 1972. 

Both the Pawley and Te Bail methods of intensity extraction or pattern decomposition have their own benefits and shortcomings. 

The two related full pattern decomposition methods in common use today were suggested by Pawley and by Le Bail et al Pawley s approach is based on Eq. 2.48 and full pattern decomposition in the case of dual wavelength data, when Kai/Ka2 doublets are present, is performed by solving the following system of equations using a least squares minimization ... 

Le Bail s approach is also based on Eq. 6.6 and it differs from the Pawley s method in that the individual intensities remain unaltered during each least squares cycle. They are extracted from a total observed intensity of the pattern between the least squares cycles after subtraction of the background. The extraction is performed using a decomposition approach that has been employed in the Rietveld method since its development, in which intensity observed in every point of the powder pattern is divided among different reflections proportionally to their calculated intensities ... 

It is worth noting that in the Le Bail s decomposition, the number of free least squares variables becomes independent of the number of Bragg reflections and only background, peak shape and lattice parameters are refined during each least squares cycle. A small inconvenience of Le Bail s approach is that the unit cell dimensions should be known with a greater precision than in Pawley s method. It also takes more least squares... 

The Bragg residual, Rb (this figure of merit is quite important in Rietveld refinement but has little to no use during full pattern decomposition because only observed Bragg intensities are meaningful in both Pawley and Le Bail methods) ... 

To a certain extent, the Rietveld method (also known as the full pattern or the full profile refinement) is similar to the full pattern decomposition using Pawley and/or Le Bail algorithms, except that the values of the integrated... 

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. 

The final step before structure determination by traditional methods is extraction of integrated intensities from the powder diffraction pattern (see below). Intensities can nearly always be determined only after deconvolution of partially overlapping Bragg reflections. This can be done for individual peaks or for groups of peaks, but usually full decomposition methods are used. There are basically two techniques the iterative Le Bail method [6] and the constrained linear least-squares approach developed by Pawley [7]. 

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