General equivalent positions

Figure If shows the molecular structure of the [Tc6Br6(/t3-Br)5]2- anion. In general, the structure of this anion is similar to the structure of the well-known octahedral halogenide clusters of molybdenum and tungsten [M6X8]4 + (X = Cl,Br, I) [5,8]. The principal difference is that the eight equivalent positions of bridging bromine atoms in the technetium clusters are not fully occupied.

N Is the number of molecules per unit volume (packing density factor), fv Is a Lorentz local field correction at frequency v(fv= [(nv)2 + 2]/3, v = u) or 2u). Although generally admitted, this type of local field correction Is an approximation vdilch certainly deserves further Investigation. IJK (resp Ijk) are axis denominations of the crystalline (resp. molecular) reference frames, n(g) Is the number of equivalent positions In the unit cell for the crystal point symmetry group g bjjj, crystalline nonlinearity per molecule, has been recently Introduced 0.4) to get general expressions, lndependant of the actual number of molecules within the unit cell (possibly a (sub) multiple of n(g)). 

For Zn, Cd, Hg, a covalent bonding contribution results in an increase of the c/a values 1.856 for Zn, 1.886 for Cd with bonds stronger in the layers than between the layers. Several binary (and complex) phases have also been described with this type of structure. These are generally solid solution phases with a random distribution of the different atomic species in the two equivalent positions. 

A graphical depiction of log(kjp) plotted as a function of log( Xa/p), where is the rate constant for general acid catalysis, is the acid ionization constant, p is the number of equivalent protons on the acid, and q is the number of equivalent positions where a proton can be... 

Example 16.2-1 List the symmetry operators of the space group 33, Pna2t. What is the point group of this space group Find the equivalent positions [x y z ] that result from applying these operators to the general point [xyz]. 

Diagrams showing (a) the general equivalent positions and (b) die symmetry elements in space group P2 /c as presented in International Tables for X-Ray Crystallography Volume I. The bold numerals in the left figure are added for die purpose of discussion. 

Consider the general equivalent positions of space group P2 /c as shown in Fig. 9.3.4(a). Let position 1 approach the origin of the unit cell in other words, let the coordinates x 0, y 0, and z 0. As this happens, position 4" also approaches the origin, while both 2 and 3 simultaneously approach the center of inversion at (0, 1 /2, 1 /2). When x = 0, y = 0, and z = 0,1 and 4 coalesce into one, and 2 and 3 likewise become the same position. There remain only two equivalent positions (0, 0, 0) and (0, 1/2, 1/2) that occupy sites of symmetry I, and they constitute the special equivalent position 2(a), which is designated as Wyckoff position 2(a). Other sets of special equivalent positions of site symmetry I are obtained by setting x = 1 /2, y = 0, z = 0 x = 0, y = 0,... 

Unique axis is b, which leads to C2/c alternative cell choices are described in other pages. Three projections of the unit cell down b, a, and c (showing the symmetry elements), and a projection down b (showing the general equivalent positions). 

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) ... 

Equivalent positions The complete set of positions produced by the operation of the symmetry elements of the space group upon any general position. 

When a point (or an atom) is placed on a finite symmetry element that converts the point into itself, the multiplicity of the site is reduced by an integer factor when compared to the multiplicity of the general site. Since different finite S5mimetry elements may be present in the same space group symmetry, the total number of different "non-general" sites (they are called special sites or special equivalent positions) may exceed one. Contrary to a general equivalent position, one, two or all three coordinates will be constrained in every atom occupying a special equivalent position. 

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). 

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