Overall atomic displacement parameter

The refinement of the overall B results in a significant reduction of all residuals (fourth line in Table 7.3), which is also expected since now we have a realistic global estimate of thermal motions of atoms in the crystal lattice of LaNi4,85Sno.i5. The refined value of the overall atomic displacement parameter is 5 = 1.32(2) A, where the number in parenthesis indicates a standard deviation in the last significant digit. 

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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. 

All of the above plus overall atomic displacement parameter 7.90 10.81 5.75 4.26... 

Overall atomic displacement parameter but individual population parameters of 2(c) and 3(g) sites plus peak shape parameters, background, zero shift, unit cell, scale 6.48 9.68 3.55 3.42... 

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.

At this point, we may begin to refine atomic displacement parameters of the individual atoms. This is done by substituting individual B s (which were kept at 0) by the refined overall atomic displacement parameter. The overall B after this substitution should be set to 0. The distribution of atoms in the model of the crystal structure of LaNi4,85Sno.i5 is such that two of the sites, 2(c) and 3(g), see Table 7.2, are occupied by different types of atoms simultaneously. Generally, atomic displacement parameters of different atoms occupying the same crystallographic site should be constrained at identical values. ... 

The presence of an impurity phase in this specimen may also be accounted for by including its crystal structure into a normal Rietveld refinement process rather than as a Le Bail s phase. For the sake of illustration, this has been done and the set of fully refined parameters can be found in the data file Ch7Ex01m.inp. It is unfeasible to refine the chemical composition of the Ni-based impurity because of its low concentration in the sample. Thus, only the scale factor, unit cell dimensions and overall atomic displacement parameter have been refined for the impurity phase, while its chemical composition was assumed as Ni4.85Sno.15. All residuals remain practically identical to those shown in the last row of Table 7.7. 

When the precision of x-ray diffraction data is high, which appears to be the case here, it is possible to refine the population of different crystallographic sites to eliminate guesses and obtain a quantitative result. The best way to do so is to return to the overall displacement parameter and instead of refining individual atomic displacement parameters, include the refinement of the individual population parameters in the corresponding sites. ... 

Visual analysis oi Figure 7.14 immediately indicates that the agreement between the observed and calculated intensity is poor. Furthermore, Rietveld refinement of this model of the crystal structure, during which the coordinates of atoms in the 4(e) site plus population of 4(d) and 4(e) sites were optimized together with the overall isotropic displacement parameter, results in the removal of Rh atoms from all sites population of both sites by Rh becomes negative (see the data file Ch7Ex02c.inp on the CD). 

When the two preferred orientation axes are assumed, the ratio between them should be refined as well. This was done subsequently, together with the refinement of the coordinates of individual atoms, overall isotropic displacement parameter, Uovei, and peak asymmetry, a. The resulting fit is substantially improved, as shown in Figure 7.20. It is clear, however, that there are still some differences between the observed and calculated intensities, as well as in the peak shapes (e.g. see the inset in Figure 7.20, where some calculated peaks appear too narrow when compared with the observed peak shapes). 

The overall isotropic atomic displacement parameter was assumed at 17iso= 0.015 A ... 

For every atom in the model that is located on a general position in the unit cell, there are three atomic coordinates and one or six atomic displacement parameters (one for isotropic, six for anisotropic models) to be refined. In addition there is one overall scale factor per structure (osf, or the first free variable in SHELXL see Section 2.7) and possibly several additional scale factors, like tbe batch scale factors in the refinement of twirmed structures, the Flack-x parameter for non-centrosymmetric structures, one parameter for extinction, etc. In addition to the overall scale factor, SHELXL allows for up to 98 additional free variables to be refined independently. These variables can be tied to site occupancy factors (see Chapter 5) and a variety of other parameters such as interatomic distances. 

Important information is included in the anisotropic atomic displacement parameters for lithium, which determine the overall anisotropy of the thermal vibration by the shape of ellipsoid. Green ellipsoids shown in Figs. 14.11a, c and 13 represent the refined lithium vibration. The preferable direction of fhennal displacement is toward the face-shared vacant tetrahedra. The expected curved one-dimensional continuous chain of lithium atoms is drawn in Fig. 14.13 and is consistent with the computational prediction by Morgan et al. [22] and Islam et al. [23]. Such anisotropic thermal vibratiOTis of lithium were further supported by the Fourier synthesis of the model-independent nuclear distribution of lithium (see Fig. 14.14). 

Scale, all profile, a and c, overall displacement plus coordinates of individual atoms and individual isotropic displacement parameters 3.97 5.24 1.89 2.91... 

The initial model of the crystal structure results in acceptable residuals without refinement of coordinates and displacement parameters of individual atoms. When the coordinates of all atoms and the overall displacement parameter were included into the least squares, the residuals further improve (row four in Table 7.14). The biggest improvement is observed in the Bragg residual, Rb, which is expected because this figure of merit is mostly affected by the adequacy of the structural model and it is least affected by the inaccuracies in profile parameters. 

Refinement of the individual isotropic parameters of all atoms yields a small negative 5 of Si 1. It is unfeasible that Nd atoms are statistically mixed in the same sites with Si because their volumes are too different ( 27 for Nd versus 1 K for Si). Given the density of the alloy, it is also impossible that all sites except this one are partially occupied. Therefore, the negative 5sii is likely due to the fact that Si atoms have only a fraction of the scattering ability of Nd atoms, and individual displacement parameters of the former cannot be reliably determined from this experiment. Another possible reason is the non-ideality of the selected peak shape function, or other small but unaccounted systematic errors. One of these is an unknown polarization constant of the employed monochromator (see Eq. 2.69). Another possibility is a more complex preferred orientation. As a result, the isotropic displacement parameters of two independent sites occupied by Si were constrained to be identical in a way, the Si atoms were refined in an overall isotropic approximation. 

Figure 7.20. The observed and calculated powder diffraction patterns of NiMn02(OH) after preferred orientation, coordinates of all atoms and the overall displacement parameter were refined in addition to the scale factor, unit cell dimensions, background, grain size and strain effects, and peak asymmetry. The insert clarifies the range between 70 and 90° 20.

Displacement parameters of oxygen and hydrogen atoms were refined in the overall isotropic approximation. 

All, plus individual isotropic displacement parameters of Gd atoms Si/Ge atoms displacements in overall approximation preferred orientation along [001]" 5.10 6.76 2.67 9.61... 

Before going into interpretation of the structural data, the overall quality of the model needs to be assessed. In addition to the standard statistics provided for models at lower resolution (Rwork. free. aii. agreement of the model with stereochemical restraints, mean B-values, etc.), quantities that characterize the atomic resolution model such as agreement of the anisotropic displacement parameters with the restraints imposed, must be quoted. Some interesting statistics on ADPs can be obtained with Ethan Merritt s PARVATI-server (Merritt, 1999). 

The shear work done for one atomic (molecular) displacement, b is the applied force times the displacement, or xb3. This work must equal the promotion energy 2Eg. Therefore, letting b3 equal the molecular volume, Vm, the required shear stress is approximately 2Eg/Vm. The parameter [Eg/Vm] is called the bond modulus. It has the dimensions of stress (energy per unit volume). The numerator is a measure of the resistance of a crystal to kink movement, while the denominator is proportional to the work done by the applied stress when a kink moves one unit distance. Overall, the bond modulus is a measure of the shear strengths of covalent bonds. 

Surface diffusion can be studied with a wide variety of methods using both macroscopic and microscopic techniques of great diversity.98 Basically three methods can be used. One measures the time dependence of the concentration profile of diffusing atoms, one the time correlation of the concentration fluctuations, or the fluctuations of the number of diffusion atoms within a specified area, and one the mean square displacement, or the second moment, of a diffusing atom. When macroscopic techniques are used to study surface diffusion, diffusion parameters are usually derived from the rate of change of the shape of a sharply structured microscopic object, or from the rate of advancement of a sharply defined boundary of an adsorption layer, produced either by using a shadowed deposition method or by fast pulsed-laser thermal desorption of an area covered with an adsorbed species. The derived diffusion parameters really describe the overall effect of many different atomic steps, such as the formation of adatoms from kink sites, ledge sites... 

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