Special position constraints

Atoms on special positions require constraints for their coordinates and sometimes also their anisotropic displacement parameters. In addition toe occupancies of atoms on special positions—and sometimes also of those atoms bound to them— need to 

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Table 2.1 Examples of special position constraints on coordinates, anisotropic displacement parameters and site occupancy factors...

Special position Constraints on coordinates Constraints on Ijy values Constraints on occupancies... 

The refinement of such disorders is relatively easy the second site of each atom can be calculated directly from the positions of the atoms of the first component by means of the symmetry operator of the special position. Therefore, it is not necessary to have two parts in the. ins file. Instead of part 1, PART 2, and PART 0, the disordered atoms are flanked with part -1 and part 0. The negative part number suppresses the generation of special position constraints, and bonds to symmetry-related atoms are excluded from the connectivity table. Moreover, the use of the second free variable is not indicated in such a case, as the ratio between the components is determined by the multiphcity of the special position. 

Molecules that are located very close to special positions, so that the symmetry would lead to chemically unreasonable arrangements, are treated the same way. In such a case the spec instruction, which generates all appropriate special position constraints for the specified atoms, may be helpful too. 

Another possible explanation for Q(2) could be a 95 5 or so disorder of Zr(2) and its ligands. Such a disorder, however, should also result in a higher U q value for Zr(2), which is not observed. Actually, the opposite is the case C/eq(Zr(l)) = 0.030, 7eq(Zr(2)) = 0.024. This difference can be explained with the special position constraints (see Section 2.5.2) that restrict the shape of the displacement ellipsoid of Zr(2) to fulfil the fourfold symmetry. This, in turn, artificially lowers the calculated value of C/eq for Zr(2). 

Therefore, the number of positional degrees of freedom is further reduced to only one independent coordinate in special positions where atoms are located on rotation or inversion crystallographic axes. Similar to both the general position and special sites on mirror planes, any special equivalent position on a rotation or inversion axis can accommodate many independent atoms (geometrical constraints are always applicable). 

The only remaining degree of freedom in this crystal structure is to refine the displacement parameters of all atoms in the anisotropic approximation (the presence of preferred orientation is quite imlikely since the used powder was spherical and we leave it up to the reader to verify its absence by trying to refine the texture using available experimental data). As noted in Chapter 2, special positions usually mandate certain relationships between the anisotropic atomic displacement parameters of the corresponding atoms. In the space group P6/mmm, the relevant constraints are as follows ... 

In their simplest form, these constraints can be used for atomic sites on special positions (e.g. x,x,x positions in cubic space groups) where there are special relationships between the individual atomic coordinates and also among the individual anisotropic thermal motion parameters (e.g. Ui i = U22 = U33 and Ui2 = Ui3 = U23 for cubic x,x,x sites). 

In some cases, constraints on the model exist, such as restrictions placed on atomic positional or thermal parameters for atoms in special positions in a crystal structure. The question was raised of the proper place for such information in the fitting process. The more certain the knowledge is, the more reasonable is its inclusion. Allowing the model more freedom to fit the data by omitting known constraints may be unwise, particularly when a variable in the model can be adjusted to account for systematic error. 

Other special positions or the combination of several special positions can lead to even lower occupancies. SHELXL recognizes atoms on or very close to special positions and automatically generates the constraints for all special positions in all space groups, which includes the reduction of the sof. 

Material Balances Material balance constraints are in the form of equalities. There are three types of such constraints fixed plant yield, fixed blends or splits, and unrestricted balances. Except in some special situations, such as planned shutdown of the plant or storage movements, the right hand-side of the balance constraints is always zero. For the purpose of consistency, flow into the plant or stream junction has negative coefficients and flows out have positive coefficients. The constraints are as follows ... 

Sources of external potential can be produced in a number of ways in which there is no need for special massive nuclei. As they identify external sources they are classical variables, namely, position coordinates for the sources. There is no quantum dynamics related to them yet. Symmetry constraints can be naturally defined. We formally write He(p a) to distinguish this situation from the standard approach. Since the primacy is given to the electronic wave function, and no Schrodinger equation is available at this point, its existence has to be taken as workinghypotheses. 

The Eulerian finite difference scheme aims to replace the wave equations which describe the acoustic response of anechoic structures with a numerical analogue. The response functions are typically approximated by series of parabolas. Material discontinuities are similarly treated unless special boundary conditions are considered. This will introduce some smearing of the solution ( ). Propagation of acoustic excitation across water-air, water-steel and elastomer-air have been computed to accuracies better than two percent error ( ). In two-dimensional calculations, errors below five percent are practicable. The position of the boundaries are in general considered to be fixed. These constraints limit the Eulerian scheme to the calculation of acoustic responses of anechoic structures without, simultaneously, considering non-acoustic pressure deformations. However, Eulerian schemes may lead to relatively simple algorithms, as evident from Equation (20), which enable multi-dimensional computations to be carried out in a reasonable time. 

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