Atomic positional parameters

In Table 2 there are given data pertinent to the lattice-parameter determination, as well as intensity data used in the determination of atomic positional parameters. 

This structure, based on the space group OjtPm in, involves about fifty atomic positional parameters, for which approximate values can be estimated from the Mg32(Al,Zn)49 parameters. Refinement of the structure would be laborious. 

In order to have around each atom in this hexagonal structure four exactly equidistant neighbouring atoms, the axial ratio should have the ideal value (8/3 that is 1.633. The experimental values range from 1.59 to 1.66. This practical constancy of the axial ratio, in contrast with what is observed for other families of isostructural compounds such as those of the NiAs type, may be attributed to a sort of rigidity of the tetrahedral (sp3) chemical bonds. As for the atomic positional parameters, the ideal value of one of the parameters (being the other one fixed at zero by conventionally shifting the origin of the cell) is z = 3/8 = 0.3750. The C diamond, sphalerite- and wurtzite-type structures are well-known examples of the normal tetrahedral structures (see 3.9.2.2). 

Table 2. Atomic positional parameters estimated from the reconstructed exit wave, those obtained after the refinement with all Cu atoms and subsequently refined occupancies...

Table 3. Final atomic position parameters for Ce5Cuj9Pj2 determined by MSLS and for La5Cuj9Pj2 determined X-ray single crystal diffraction data.

The goodness of fit achieved in a refinement is defined by Eq. (4.27). Its evaluation for each of the subsets of data requires partitioning the unknowns between the two sets. Some, such as the scale factors, will be dependent on one subset only other, such as heavy-atom positional parameters, will be determined by both sets of data. Information on the relative dependency is contained in the matrix B, the elements of which are the sums of the contributions from the X-ray and neutron observations. We may write... 

Let us suppose that the atomic positional parameters r, and the thermal smearing factors 71 have been obtained by neutron diffraction. We can then calculate the X-ray scattering of the procrystal density with spherical-atom X-ray scattering factors, using... 

While this has long been the most popular method to study crystal structures, it was not until the introduction of ADXRD on synchrotron sources in the 1990s that it became routine to determine atomic position parameters using this technique at high pressure. While such studies have been performed principally at synchrotron sources, laboratory-based studies have also been performed see for example [155-158]. 

Table 2 Atomic Positional Parameters for BaPb03, space group Ibmm...

Table 3 Atomic Positional Parameters for BaPbjsBi 25O3...

Finally, it has to be mentioned that, in addition to the magnetically induced effects on the lattice parameters, there is also a shifting of the Gd atoms within the unit cell. As discussed above, the relative atomic position parameter ZGd changes by about —0.003 due to the magnetic ordering. 

The number of parameters adjusted was generally in the range 10-30 (7 to 51 in the x-ray cases reviewed by Young (6.)). The standard deviations in atom position parameters are typically 1-4 parts in 1,000 in the x-ray cases reviewed. Fig. 3 shows the x-ray patterns for a fluorapatite, CasfPOOsF, case in which 28 parameters were varied (20 of them structural), Rp= 12.1%, and RWn= 13.9% (4.). In this Fig., the observed pattern is indicated by the data points with vertical error bars, the calculated pattern is the continuous curve overlying them, the difference between observed and calculated patterns is shown by the lower curve, and the positions of Bragg peaks are shown by the vertical bars between the upper and lower curves. In this case the fit both looks to be very good and is so indicated by the R value. 

All atomic positional parameters, which define the crystal and molecular structure, can be assigned to the general and special equivalent positions of space group Fd3 (origin at 23) ... 

A preliminary knowledge of the crystal structure is important prior to a detailed charge density analysis. Direct methods are commonly used to solve structures in the spherical atom approximation. The most popular code is the Shelx from Sheldrick [26] which provides excellent graphical tools for visualization. The refinement of the atom positional parameters and anisotropic temperature factors are carried out by applying the full-matrix least-squares method on a data corrected if found necessary, for absorption and diffuse scattering. Hydrogen atoms are either fixed at idealized positions or located using the difference Fourier technique. 

Single crystals, suitable for X-ray crystallography, were obtained directly from the synthesis of the compound (solvent = anhydrous methylene chloride). The crystal structure was determined on a CAD-4 ENRAF-NONIUS PDP 8/M computer-controlled single-crystal diffractometer. Atomic positional parameters, bond lengths and bond angles are published elsewhere 33 >. 

The form of this bibliography is similar to that of previous years.1-3 As for that for 1974, projection diagrams of the carbohydrate, nucleoside, or nucleotide molecules or ions have been provided, where possible. The diagrams were produced directly from the unit-cell and atomic positional parameters given in the papers. In a few cases, corrections were necessary. 

Although correlation between parameters is a function of the data structure and has nothing to do with deficiencies in the model, it has implications for both the choice of the model and the design of the experiment. EVANS described his experiences with the determination of the crystal structure of tetragonal barium titanate (BaTiOa). The problem was ample in that it involved only three atomic positional parameters (one for Ti and two for 0), plus nine thermal parameters. There was considerable interest in the details of the structure because of the ferroelectric properties of the material. The proposed model was essentially a simple cubic arrangement of atoms, but with Ti displaced slightly from the center of an octahedron. By ordinary x-ray standards, this distortion (which was expected to be on the order of 0.15 A) could be measured with a standard error of 0.01-0.02 A if... 

Fig. 30. Schematic representation of the L2, structure type. The atomic position parameters are A(0 0 0),...

NajNbO have been recorded and interpreted on the basis of a rock salt lattice containing a disordered distribution of Na and Nb atoms among the cation sites. A corrected atomic position parameter for Ba3Si4Nbg026 has been given. ... 

Once least-squares methods came into general use it became standard practice to refine not only atomic positional parameters but also the anisotropic thermal parameters or displacement parameters (ADPs), as they are now called [22]. These quantities are calculated routinely for thousands of crystal structures each year, but they do not always get the attention they merit. It is true that much of the ADP information is of poor quality, but it is also true that ADPs from reasonably careful routine analyses based on modem point-by-point or area diffractometer measurements can yield physically significant information about atomic motions in solids. We may tend to think of crystal structures as static, but in reality the molecules undergo translational and rotational vibrations about their equilibrium positions and orientations, as well as internal motions. Cruickshank taught us in 1956 how analysis of ADPs can yield information about the molecular rigid-body motion [23], and many improvements and modifications have been introduced since then. In particular, various computer programs are available to estimate the amplitudes of simple postulated types of internal molecular motion e.g., torsion-... 

R2Ni22C3 compound has an orthorhombic structure with space group Cmca and eight formula units per unit cell. The strueture was solved by Patterson and Fourier methods and refined by a least-squares caleulation to R = 0.082. The atomic position parameters have been determined (Bodak et al. 1982). The lattiee parameters of the compounds La2Ni22C3, Ce2Ni22C3 and Pr2Ni22C3 11.45(5), 11.384(3)... 

From a crystallographic perspective, the ideal perovskite stmeture is inflexible, as the unit cell has no adjustable atomic position parameters, so that any compositional change must be accommodated by a change in lattice parameter. This is a... 

Atom Positional parameters y 2 Unit cell dimensions (A) a b c ... 

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