Equivalent positions in space groups

We have stated that the arrangement of faces on a crystal (or more accurately the arrangement of face-normals) can be described by its point symmetry. The six 

Variation in habit of crystals. Different relative development of (a) cube faces, and (b) cube and octahedron faces. 

Relation between hexagonal and rhombohedral unit cells. 

The crystal systems unit cells and characteristic symmetry 

System Relations between edges and angles of unit cell Lengths and angles to be specified Characteristic symmetry 

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Fig. 155. Symmetry Moments and equivalent positions in space-group Cmca.

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

The positions of the four oxygen atoms can best be deduced by referring to the lists of equivalent positions in International Tables (1952). There are five sets of fourfold positions in space-group P42 mnm ... 

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,... 

There are other circumstances in which some of the atomic coordinates in a crystal can be discovered by evaluation of the Patterson function over a particular plane or along a particular line. For instance, it may be known, from a consideration of the space-group and the equivalent positions in the unit cell, that there is one particular atom at the origin of the cell and others somewhere on the plane y = L The... 

Patterson function will show maxima on this plane at positions which give immediately the actual coordinates of these atoms. Similar considerations were used in the determination of the structure of potassium sulphamate NH2SO3K (Brown and Cox, 1940) it was known that the y coordinates of the potassium ions are 0 and while those of the sulphur atoms are and consequently, the Patterson function on the plane y = l shows maxima at positions corresponding to K-S vectors. Atomic positions are not given directly, but can be derived from the positions of Patterson peaks by a consideration of the equivalent positions in the space-group. 

There are occasionally more subtle traps. Some arise out of faulty reasoning and are thus, in principle, entirely avoidable. If, for example, a crystal is found to contain a number of molecules equal to the multiplicity of the general position for its space group, that does not necessarily mean that the molecules must be on general positions. Consider the simplest possible case, a crystal in space group PI with two molecules in the unit cell. While in most instances there are two equivalent molecules on the twofold general position, cases are also known in which each molecule resides on a different center of inversion. In every such case this is a logical possibility and only an actual structure solution can rule it in or out. 

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. 

To describe the contents of a unit cell, it is sufficient to specify the coordinates of only one atom in each equivalent set of atoms, since the other atomic positions in the set are readily deduced from space group symmetry. The collection of symmetry-independent atoms in the unit cell is called the asymmetric unit of the crystal structure. In the International Tables, a portion of the unit cell (and hence its contents) is designated as the asymmetric unit. For instance, in space group P2 /c, a quarter of the unit cell within the boundaries 0asymmetric unit. Note that the asymmetric unit may be chosen in different ways in practice, it is preferable to choose independent atoms that are connected to form a complete molecule or a molecular fragment. It is also advisable, whenever possible, to take atoms whose fractional coordinates are positive and lie within or close to the octant 0 < x < 1/2,0 < y < 1/2, and 0 < z < 1 /2. Note also that if a molecule constitutes the asymmetric unit, its component atoms may be related by non-crystallographic symmetry. In other words, the symmetry of the site at which the molecule is located may be a subgroup of the idealized molecular point group. 

Table 10.3.1. Atomic coordinates of equivalent positions in cubic space group O — ImSiir...

The nondegenerate geminal pairs are usually named according to their chemical shifts (e.g., downfield of ft ) rather than their stereochemical relationships (pro-R and pro-S). In structure calculations, this usually is dealt with by creating a pseudo-atom right between the pro-R and pro-S positions in 3D space. The NOE restraints are applied to the pseudoatom and not to the real atoms, and the distance limit is increased a bit to account for the ambiguity (we do not really know which restraint applies to which of the two positions in space). Similarly, a pseudoatom is created at the center of the three equivalent protons of a CH3 group, and the distance restraint is applied to the pseudoatom. 

Table 2.3 Wyckoff Positions (WP) for symmorphic space group 187 (P-6m2/ Djj ) with the point group Dyj, as the factor group G/T, with T the group of translations. Note, especially that the symbols x, y and z in the tables are the magnitudes along the a, b and c edges of the hexagonal unit cell. The entries in the column SG identify the site groups for the different sets of equivalent positions in the unit cells distinguished by the different WP.

The eight equivalent positions in the space group Pbca are ... 

The structure of caesium phosphide, Cs3P7, belongs in space group P41 and has lattice parameters a = b = 0.9046 nm, c = 1.6714 nm. The structure is tetragonal, and the screw axis runs parallel to the c-axis. All of the atoms must be placed in equivalent general positions, the... 

The three Harker sections are shown. The largest nonorigin peaks yield the coordinates of two independent gold sites, Aul and Au2, at (x,y,z) values of (0.2293, 0.0792, 0.2291) and (0.4969, 0.2228, 0.1428). Vectors between Aul and Au2 occur on general sections (not shown here). The origin peak is 1000 and contours start at 25 and increase in intervals of 25. The equivalent positions in the space group... 

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