Systematics lattice parameters

Thus the rather easily obtained atomic sizes are the best indicator of what the f-electrons are doing. It has been noted that for all metallic compounds in the literature where an f-band is believed not to occur, that the lanthanide and actinide lattice parameters appear to be identical within experimental error (12). This actually raises the question as to why the lanthanide and actinide contractions (no f-bands) for the pure elements are different. Analogies to the compounds and to the identical sizes of the 4d- and 5d- electron metals would suggest otherwise. The useful point here is that since the 4f- and 5f-compounds have the same lattice parameters when f-bands are not present, it simplifies following the systematics and clearly demonstrates that actinides are worthy of that name. 

It has been noted that the conductivity and activation energy can be correlated with the ionic radius of the dopant ions, with a minimum in activation energy occurring for those dopants whose radius most closely matches that of Ce4+. Kilner et al. [83] suggested that it would be more appropriate to evaluate the relative ion mismatch of dopant and host by comparing the cubic lattice parameter of the relevant rare-earth oxide. Kim [84] extended this approach by a systematic analysis of the effect of dopant ionic radius upon the relevant host lattice and gave the following empirical relation between the lattice constant of doped-ceria solid solutions and the ionic radius of the dopants. 

Gschneidner Jr., K.A., and Calderwood, EW. (1986) Intra-rare earth binary alloys phase relationships, lattice parameters and systematics. In Handbook on the Physics and Chemistry of Rare Earths, eds. Gschneidner Jr., K.A. and Eyring, L. (North-Holland, Amsterdam), Vol. 8, p. 1. 

For nanocrystals, the interpretation of lattice parameter shifts is complicated by the very small dimensions of the crystallites. Because of the small crystal dimensions, the diffraction peaks are broadened as described by the Debye-Scherrer equation (106), making accurate assessment of small shifts more challenging. Systematic errors such as zero-point or sample-height offsets can also cause artificial shifts in lattice constants (107). The inclusion of an internal... 

K.A. Gschneidner Jr and F.W. Calderwood, Intra rare earth binary alloys phase relationships, lattice parameters and systematics 1... 

When the A ion is kept constant and the B ion is varied, systematics in lattice parameters are not obvious. In Fig. 9, the lattice parameters and da ratios (35, 36) of some Sr2B04 compounds are plotted against the ionic radius of the B ion. The a parameter varies linearly with the radius of the B ion provided that it has partially filled d orbitals. Thus, ions such as Sn4+, Hf4+, and Zr4+ do not fall on this straight line. Poix (3) has, however, found a linear relationship using the /3B parameters. What is important is that there is no linear relationship between the c parameters or the da ratios and the size of the B ion in these compounds. Furthermore, compounds containing B ions with partially filled d orbitals exhibit larger da ratios than those with filled or empty d orbitals. When the B ions have partially filled d orbitals, the da ratio... 

The mean systematic deviation between the observed and the calculated values of the unit cell s mass—or, what amounts to the same thing, the density-increases as S/Ti decreases. Now, the sulfides belonging to the TiS2 and Ti2S3 phases are always well crystallized, and the errors in measuring lattice parameters and densities are of the same order. It is thus not unreasonable that the increase of this deviation should be due to the creation of sulfur vacancies, which proceeds simultaneously with the insertion of titanium. The observation has been previously made in connection with other analogous systems in which there is a transition from the compound MX2 to MX. 

Berger, H. Systematic errors in precision lattice-parameter determination of single crystals caused by asymmetric line profiles. J. Appl. Cryst. 19, 34-38 (1986). 

Regardless of which type of preferred orientation is present in the sample, it will in its own and systematic way affect diffracted intensities. In severe cases, nothing more than lattice parameters (if any) can be determined from highly textured powder diffraction data since it is impossible to precisely account for the changes in the diffracted intensity caused by the exceedingly non-random distribution of particle orientations. 

Least squares refinement of lattice parameter (Eq. 5.39) assuming unit weights and using all 20 available Bragg peaks results in a = 4.1599(3) A. The obtained differences between the observed and calculated 26 are shown in Figure 5.19 and it is quite obvious that there is a systematic dependence of A20 on the Bragg angle. A similar behavior is always indicative of a systematic error, namely the presence of zero shift or sample displacement errors, or a combination of both. 

Figure 5.19. The differences between the observed and calculated Bragg angles after least squares refinement of the lattice parameter of LaBs without accounting for the presence of any kind of systematic error (open circles) using a = 4.1599(3) A. The dash-dotted line drawn through the data points is a guide for the eye. The solid line represents corrections of the observed Bragg angles using the refined in the next step sample displacement error (s/R = 0.00632) and the dashed line represents a similar correction by using the determined zero shift error (50o = 0.078°).

However, if one compares the values of the lattice parameter obtained when a different kind of a systematic error was assumed and accounted for in the data, the difference between the two is statistically significant (4.1583 vs. 4.1574 A for sample displacement and zero shift effects, respectively). This is expected given the different contribution from different errors as seen in Figure 5.19. Usually, both effects are present in experimental data. The refinement of two contributions simultaneously is, however, not feasible due to strong correlations between sample displacement and zero shift parameters as shown in Figure 5.21. 

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