General and special positions

In the previous section, an atom or a motif was allocated to a position, (x, y), which was anywhere in the unit cell. Such a position is called a general position, and the other positions generated by the symmetry operators present are called general equivalent positions. The number of general equivalent positions is given by the multiplicity. Thus, the general position (x, y) in the plane group p2, (number 2) has a multiplicity 

however, the x, y) position of an atom falls upon a symmetry element, the multiplicity will decrease. For example, an atom placed at the unit cell origin in the group p2, will lie on the diad axis, and will not be repeated. This position will then have a multiplicity of 1, whereas a general point has a multiplicity of 2. Such a position is called special position. The special positions in a unit cell conforming to the plane group p2 are found at coordinates (0, 0), Q/i, 0), (0, Vi) and (Vi, Vi), coinciding with the diad axes. There are four such special positions, each with a multiplicity of 1. 

Note that the multiplicity of a diad in plane group p2 is different from that of a diad in plane group p4. The multiplicity of a special position 

Both the general and special positions in a unit cell are generally listed in a table that gives the multiplicity of each position, the 

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Table 9.3.9. General and special positions of space group P2 /c.

A set of symmetry-equivalent coordinates is said to be a special position if each point is mapped onto itself by one other symmetry operation of the space group. In the space group Pmmm, there are six unique special positions, each with a multiplicity of four, and 12 unique special positions, each with a multiplicity of two. If the center of a molecule happens to reside at a special position, the molecule must have at least as high a symmetry as the site symmetry of the special position. Both general and special positions are also called Wyckoff... 

Both general and special positions are also called Wyckoff positions, in honor of the American crystallographer Ralph Walter Graystone Wyckoff (1897-1994). Wyckoff s 1922 book. The Analytical Expression of the Results of the Theory of Space Groups, contained tables with the general and special positional coordinates permitted by the symmetry elements. This book was the foremnner of International Tables for X-ray Crystallography, which first appeared in 1935. 

Ralph Walter Graystone Wyckoff (1897-1994) publishes The Analytical Expression of the Results of the Theory of Space Groups, containing positional coordinates for general and special positions permitted by the symmetry elements. 

All the general and special positions for all of the plane groups are listed in the International Tables for Crystallography. 

Determine the general and special positions (orbits) along with the symmetry elements for the plane group p4gm. 

Decoration of the Lattice with the Basis At this point we have to recall that in a real crystal structure we have not only the lattice, but also the basis. In [3.8], there are standardized sets of general and special positions (i. e. coordinates x, y, z) within the unit cell (Wyckoff positions). An atom placed in a general position is transformed into more than one atom by the action of all symmetry operators of the... 

Deep blue crystals of cuprammonium sulfate [Cu(NH3)4]S04 H2O, are orthorhombic with a unit cell of dimensions a = 7.08,b = 12.14,c= 10.68A,andZ = 4.Thespace group has the following sets of general and special 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) ... 

With alloys and substitutional solid solutions, it is possible that a mixture of atoms (of similar size, valence, etc.) may reside at a general or special position and all its equivalent coordinates. The fraction of atoms of one type residing at that position is given by the site occupancy, or site occupation factor. The sum of the site occupation factors for that site must equal unity. The distribution of two or more types of atoms over a single site is completely random. Where two atoms are distributed over all the equivalent coordinates of different sites with similar local coordination environments (but not identical site symmetry), electronic, or other, effects can result in partial site preferences. That is, there can be a nonstatistical distribution over the two sites. 

What is the difference between a crystallographic general position and special position ... 

A set of points that is equivalent with respect to a symmetry group is called an orbit. The polyhedra of Fig. 2.17 represent orbits of point groups. The arrows in Fig. 2.29 represent the orbits of plane groups. For the majority of groups, there are several types of orbit that we refer to as general positions and special positions. We will illustrate this important point with the aid of the plane group p2mg (Fig, 2.30). 

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

These include rotation axes of orders two, tliree, four and six and mirror planes. They also include screM/ axes, in which a rotation operation is combined witii a translation parallel to the rotation axis in such a way that repeated application becomes a translation of the lattice, and glide planes, where a mirror reflection is combined with a translation parallel to the plane of half of a lattice translation. Each space group has a general position in which the tln-ee position coordinates, x, y and z, are independent, and most also have special positions, in which one or more coordinates are either fixed or constrained to be linear fimctions of other coordinates. The properties of the space groups are tabulated in the International Tables for Crystallography vol A [21]. 

In a general case of a mixture, no component takes preference and the standard state is that of the pure component. In solutions, however, one component, termed the solvent, is treated differently from the others, called solutes. Dilute solutions occupy a special position, as the solvent is present in a large excess. The quantities pertaining to the solvent are denoted by the subscript 0 and those of the solute by the subscript 1. For >0 and x0-+ 1, Po = Po and P — kxxx. Equation (1.1.5) is again valid for the chemical potentials of both components. The standard chemical potential of the solvent is defined in the same way as the standard chemical potential of the component of an ideal mixture, the standard state being that of the pure solvent. The standard chemical potential of the dissolved component jU is the chemical potential of that pure component in the physically unattainable state corresponding to linear extrapolation of the behaviour of this component according to Henry s law up to point xx = 1 at the temperature of the mixture T and at pressure p = kx, which is the proportionality constant of Henry s law. 

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