Wyckoff position

The group-subgroup relation of the symmetry reduction from diamond to zinc blende is shown in Fig. 18.3. Some comments concerning the terminology have been included. In both structures the atoms have identical coordinates and site symmetries. The unit cell of diamond contains eight C atoms in symmetry-equivalent positions (Wyckoff position 8a). With the symmetry reduction the atomic positions split to two independent positions (4a and 4c) which are occupied in zinc blende by zinc and sulfur atoms. The space groups are translationengleiche the dimensions of the unit cells correspond to each other. The index of the symmetry reduction is 2 exactly half of all symmetry operations is lost. This includes the inversion centers which in diamond are present in the centers of the C-C bonds. 

In these structural types the atoms are distributed in three groups of positions corresponding (obviously with different values of the x and z free parameters) to the same type of Wyckofif positions (Wyckoff position c). 

Number of positions, Wyckoff notation, and point symmetry 2 i 1 x,y,z ... 

Number of positions, WyckofF notation and point symmetry... 

No. of positions Wyckoff notation Site symmetry Coordinates of equivalent positions... 

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 ... 

Element symbol, atom identifier for this atom, oxidation state, number of positions, Wyckoff notation, atomic coordinates xyz, isotropic or anisotropic displacement factors, site occupation, all values with s.u. (standard uncertainty) Conditions of measurement (by defined acronyms) ... 

The different sets of positions in crystals are called Wyckoff positions. They are listed for every space-group type in International Tables for Crystallography, Volume A, in the following way (example space-group type Nr. 87, 74/m) ... 

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. 

A (crystallographic) orbit is the set of all points that are symmetry equivalent to a point. An orbit can be designated by the coordinate triplet of any of its points. If the coordinates of a point are fixed by symmetry, for example 0, q, then the orbit and the Wyckoff position are identical. However, if there is a free variable, for example z in 0, , z, the Wyckoff position comprises an infinity of orbits. Take the points 0, 0.2478... 

Wyckoff position 8 of the space group I4/m. Each of these points belongs to an orbit consisting of an infinity of points (don t get irritated by the singular form of the words Wyckoff position and orbit ). 

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 symmetry reduction can be continued. A (non-maximal) subgroup of F43m is I42d with doubled lattice parameter c. On the way F43m —174 2d the Wyckoff position of the zinc atoms splits once more and can be occupied by atoms of two different elements. 

The symmetry reduction to the mentioned hettotypes of diamond is necessary to allow the substitution of the C atoms by atoms of different elements. No splitting of Wyckoff positions, but a reduction of site symmetries in necessary to account for distortions of a structure. Let us consider once more MnP as a distorted variant of the nickel arsenide type (Fig. 17.5, p. 197). Fig. 18.4 shows the relations together with images of the structures. 

The aristotype can be considered to be either the packing of spheres itself, or the NiAs type which corresponds to the packing in which all octahedral interstices are occupied by Ni atoms (Wyckoff position 2a). In the aristotype these interstices are symmetry equivalent subgroups result if the interstices are occupied only partially or by different kinds of atoms (or if the Ni atoms of NiAs are partially removed or substituted). By this procedure the sites of the interstices become non-equivalent. 

Group-subgroup relations from hexagonal closest-packing of spheres to some MX3 and M2X3 structures. The boxes represent octahedral voids, with the coordinates as given at the top left. The positions of the octahedron centers are labeled by their Wyckoff letters. Gray boxes refer to occupied voids. The dots indicate how the atoms Ru, P and N are shifted from the octahedron centers parallel to c... 

Set up the Barnighausen tree for the relation between disordered and ordered AuCu3, including the relations between the Wyckoff positions (Fig. 15.1, p. 158). You will need International Tables for Crystallography [48], Volume A, and advantageously also Volume A1 [181]. Will ordered AuCu3 form twins ... 

With an increase in size of the active metals, the interlayer interstitials between the triacontahedral and the penultimate icosidodecahedral shells appear to be occupied by smaller electronegative components, with variable occupancies. These interlayer interstitials are actually the centers of cubes and correspond to the Wyckoff 8c (1/4 V4 A) special position in 1/1 ACs. Strictly speaking, occupation at this site means that the structure is no longer YCd6-type but, for convenience, they are still referred to as Tsai-type phases. According to Piao and coworkers [94], occupation of these cube centers has strong correlation with the orientations of the innermost tetrahedra and distortions of the dodecahedra. 

Atomic Positions Using Cartesian Coordinates and Corresponding Wyckoff Letters... 

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

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. 

Notice that for this structure type, examples are known of compounds, such as Y3A1 and YA13, in which the same elements alternatively occupy either the (a) or the (c) Wyckoff positions and exchange their roles in the structure. 

Considering Au in 0, 0, 0 as the reference atom, the next neighbours Au atoms are the six Au shown in Fig. 3.29(a), corresponding to the same Wyckoff position and having, in comparison with the reference atom, the coordinates 0, 0, 1 0, 0, 1 0, 1, 0 0, 1, 0 1, 0, 0 1, 0, 0, all at a distance d = a = 374.8 pm, that is at a reduced distance dr = d/dmin = 1.414. Notice that in the analysis of the structure it may be necessary to consider not only the positions of the atoms in the reference cell but also those in the adjacent cells. Notice also that, in a simple cubic structure without free positional parameters such as the AuCu3 type, the reduced distances are independent of the values of the lattice parameters and are the same for all the isostructural compounds. 

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