Equivalent position, special

Surprinslngly, we observe an drastic effect of the concentration on the SRO contribution (figure 2) indeed, in PtaV, the maxima are no longer located at a special point of the fee lattice but the (100) intensity is splltted perpendicularly in the (010) direction and presents a saddle point at (100) position. Notice that these two maxima are not located just above Bragg peaks of the ordered state the A B ground state presents Bragg peaks at ( 00) and equivalent positions whereas the SRO maxima peak between ( 00) and (100). 

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,... 

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) ... 

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). 

Two points should be emphasized. First, according to classical structure theory, all the equivalent positions of a given set should be occupied and moreover they should all be occupied by atoms of the same kind. In later chapters we shall note examples of crystals in which one or both of these criteria are not satisfied an obvious case is a solid solution in which atoms of different elements occupy at random one or more sets of equivalent positions. (The occupation of different sets of equivalent positions by atoms of the same kind occurs frequently and may lead to quite different environments of chemically similar atoms. Examples include the numerous crystals in which there is both tetrahedral and octahedral coordination of atoms of the same element—in the same oxidation state—as noted in Chapter 5, and crystals in which there is both coplanar and tetrahedral coordination of Cu(ii), p. 890, or Ni(ii), p. 965.) The second point for emphasis is if a molecule (or complex ion) is situated at one of the special positions it should possess the point symmetry of that position. A molecule lying on a plane of symmetry must itself possess a plane of symmetry, and one having its centre at the intersection of two planes of symmetry must itself possess two perpendicular planes of symmetry. If, therefore, it can be demonstrated that a molecule lies at such a position as, for example, would be the case if the unit cell of Fig. 2.13 contained only one molecule, (a fact deducible from the density of the crystal) this would constitute a proof of the symmetry of the molecule. Such a conclusion is not, of course, valid if there is any question of random orientation or free rotation of the molecules. Moreover, there is another reason for caution in applying this type of argument to inorganic crystals. 

Fig. 2.12. Conformational map of diphenylmethane. The 16 equivalent positions (open circles) are images of the 16 isometric conformations with different values of the two torsion angles. The unit cell shown is non-primitive, the primitive iattice having translation distances of n along Wa d along Wg. The plane group is cmm, with translation vectors y, = coa +Wb> 2 = Wa-Wb- The general positions of this plane group are images of arbitrary conformations, the special positions images of conformations with point-group symmetry...

A special case is represented when compounds have equivalent substitution sites, as often happens for disubstituted compounds on a phenyl ring where the second substituent may sit at two equivalent ortho or meta positions. It is clear that under these circumstances the number of possible candidates decreases for taking into account equivalent structures. This problem is handled automatically by appropriate software, as, for instance, DESDOP [30], where equivalent positions are correctly defined in the problem formulation. 

A chirality classification of crystal structures that distinguishes between homochiral (type A), heterochiral (type B), and achiral (type C) lattice types has been provided by Zorkii, Razumaeva, and Belsky [11] and expounded by Mason [12], In the type A structure, the molecules occupy a homochiral system, or a system of equivalent lattice positions. Secondary symmetry elements (e.g., inversion centers, mirror or glide planes, or higher-order inversion axes) are precluded in type A lattices. In the racemic type B lattice, the molecules occupy heterochiral systems of equivalent positions, and opposite enantiomers are related by secondary lattice symmetry operations. In type C structures, the molecules occupy achiral systems of equivalent positions, and each molecule is located on an inversion center, on a mirror plane, or on a special position of a higher-order inversion axis. If there are two or more independent sets of equivalent positions in a crystal lattice, the type D lattice becomes feasible. This structure consists of one set of type B and another of type C, but it is rare. Of the 5,000 crystal structures studied, 28.4% belong to type A, 55.6% are of type B, 15.7% belong to type C, and only 0.3% are considered as type D. 

The general and special equivalent positions in space group C2/m are ... 

In the International Tables of Crystallography, for each of the 230 space groups the list of all the Wyckoff positions is reported. For each of the positions (the general and the special ones) the coordinate triplets of the equivalent points are also given. The different positions are coded by means of the Wyckoff letter, a, b, c, etc., starting with a for the position with the lowest multiplicity and continuing in alphabetical order up to the general position. 

Alternatively, it is possible to add a functional group (FGA), either to functionalise the carbon skeleton or to create new consonances (dissonances must be always avoided ) which can provide valid bond disconnection mechanisms (HP-4). Of some special interest is the introduction of a double bond (in the a,P-position if a carbonyl group is already present in the target molecule) since it is a typical ambivalent group, of type A, which provides different valid bond disconnection mechanisms, either directly or after substitution by an equivalent synthon, such as a... 

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