Wyckoff symbol

The Wyckoff symbol is a short designation it consists of a numeral followed by a letter, for example 8/. The cipher 8 states the multiplicity, that is, the number of symmetry-equivalent points in the unit cell. The / is an alphabetical label (a, b,c,...) according to the sequence of the listing of the positions a is always the position with the highest site symmetry. 

Every space group listed in the family tree corresponds to a structure. Since the space group symbol itself states only symmetry, and gives no information about the atomic positions, additional information concerning these is necessary for every member of the family tree (Wyckoff symbol, site symmetry, atomic coordinates). The value of information of a tree is rather restricted without these data. In simple cases the data can be included in the family tree in more complicated cases an additional table is convenient. The following examples show how specifications can be made for the site occupations. Because they are more informative, it is advisable to label the space groups with their full Hermann-Mauguin symbols. 

The two special (a) and (b) Wyckoff positions have no free coordinate parameter. The two occupancy parameters are 100%. The Pearson symbol is cP2. 

Pearson symbol and prototype Space group symbol and number Lattice parameters (pm) Wyckoff positions (WP) Representative coordinates x,y, z of the 1st WP Occupancy Coordination polyhedron code Average polyhedron radius (pm) AET... 

With reference to the aforementioned structural unit slab the Jagodzinski-WyckofF symbol of the two structures is ZnS sphalerite c ZnS wurtzite h. 

The symmetry elements that intersect at a Wyckoff position determine its site symmetry. For example, a Wyckoff position that lies on the intersection of two mirror planes has mm2 (C2v) symmetry (Fig. 10.6(a)) while one that lies at the intersection of a mirror plane and a three-fold rotation axis along its normal has 3/m (Csh) symmetry (Fig. 10.6(b)). An atom lying on a general position has no symmetry other than a one-fold axis which is represented by the symbol 1 (Ci). 

This list is reproduced exactly as it appears in the International Tables. It tells us all the different kinds of locations that exist within one unit cell. In each instance we are given the multiplicity of the type of point, namely, how many of them there are that are equivalent and obtainable from each other by application of symmetry operations. There is also an italic letter, called the Wyckoff letter. This is simply an arbitrary code letter that some crystallographers sometimes find useful these letters need not concern us further. Next there is the symbol for the point symmetry that prevails at the site. Finally, there is a list of the fractional coordinates for each point in the set. 

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.

Atom Wyckoff symbol Atomic coordinates Occupation... 

From the forgoing, it is seen that in order to specify a two-dimensional crystal structure, all that needs to be specified are (i), the lattice parameters, (ii), the plane group symbol, (iii), a list of atom types together with the Wyckoff letter and the atomic coordinates. For complex crystals, it simplifies the amount of information to be recorded enormously. 

In previous discussions of crystal structures, each atom was considered to occupy a crystallographic position completely. For example, in the crystal structure of Cs3P7, described in Chapter 5, the Csl atoms occupied completely all of the positions with a Wyckoff symbol 4a. There are four equivalent Csl atoms in the unit cell. In such (normal) cases, the occupancy of the positions is said to be 1.0. 

From the symbol Pnma, we deduce that the crystal class is mmm and that the crystal system is orthorhombic. The cell angles are thus a = jS = 7 = 90°. Because Z = 4, the unit cell contains 12 atoms of Fe and 4 of C. From the International Tables we learn that the multiplicity of a general position (Wyckoff symbol d) is 8. There are three special positions a, b and c with a multiplicity of 4 ... 

The difference between oriented site-symmetry groups of different Wyckoff positions is due to different orientations of the elements of the site-symmetry group G, with respect to the lattice. The difference arises when similar symmetry elements (reflections in planes and rotations about twofold axes of symmetry) occur in more than one class of elements of the point group F. Only eleven site groups [C2(2), Cs m), C2h 2./m), C 2 (2mm), CsyiZmm), 2(222), Ds(322), D2d(42m), D3d 32m), D hijnrnm), and >3 (62m)] can have different orientations with respect to the Bra-vais lattice. Oriented site- symmetry symbols show how the symmetry elements at a site are related to the symmetry elements of a space group. The site-symmetry... 

Prototype Pearsons symbol Strukturbericht designation Space group Wyckoff positions Equivalent description... 

The first two columns of the table contain the labels of the induced reps in the q-basis (these labels number the rows of the table) the international symbols (Roman letters a, b, c and so on) of the Wyckoff positions (sites in direct space) and the Mulliken symbols of the irrep>s of the site-symmetry groups for these Wyckoff positions. For example, d a2u) and d(e ) are the labels of induced reps in q-basis for space... 

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