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General and special equivalent 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) ... [Pg.336]

The general and special equivalent positions in space group C2/m are ... [Pg.249]

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,... [Pg.322]

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... [Pg.26]

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). [Pg.68]

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. [Pg.103]

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. [Pg.23]

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). [Pg.71]

In summary, the model allows for two types of interactions between the mirror spaces, the weak kinematical perturbation and the adiabatic and sudden limits equivalent to Eq. (17) or Eqs. (29)-(34). The overwhelming rate of particles over antiparticles in the Universe is inferred in this picture once the particular particle state has been selected. The Minkowski metric of the special theory of relativity is represented here by a non-positive definite metric, Eq. (8), bringing about a quantum model with a complex symmetric ansatz. Although the latter permits general symmetry violations, it is nevertheless surprising that fundamental transformations between complex symmetric representations and canonical forms come out unitary. [Pg.131]

Before the similar situation in cylindrite can be treated, we have to describe the special type of layer stacking disorder which is inherent to non-commensurate layer structures. When two types of layers are semi-commensurate (or nearly so) and non-modulated in the semi-commensurate intralayer direction (e.g. b), the layer B can be placed on layer A in a number of ways with equivalent layer match, mutually displaced by the translation mb for the primitive mesh A, or mbAl2 + c tl in the more usual case of a centred layer mesh m stays within the range of the vernier repeat (n) of the layer A. Several of these positions will not coincide with each other, depending on the ratio of to ng in the semi-commensurate direction The same will in general be true for the positions of layer A on layer B, but this time the positions will be determined by the vectors of the layer B. [Pg.147]

Figure 7.19 Illustration of the generation process for a hypothetical framework, denoted as HI. The pore radius is set at 6.0 A. Two unique atoms are placed, one at the general position I and the other at the special position j. The first unique atom T1 is constrained to the pore wall, and the position on the wall is randomly selected. Then its 23 equivalent atoms are generated using the symmetry operation [Figure 7.19(a)]. The distance between any two atoms is calculated. Reproduced with permission from [37], Copyright (2003) American Chemical Society... Figure 7.19 Illustration of the generation process for a hypothetical framework, denoted as HI. The pore radius is set at 6.0 A. Two unique atoms are placed, one at the general position I and the other at the special position j. The first unique atom T1 is constrained to the pore wall, and the position on the wall is randomly selected. Then its 23 equivalent atoms are generated using the symmetry operation [Figure 7.19(a)]. The distance between any two atoms is calculated. Reproduced with permission from [37], Copyright (2003) American Chemical Society...
It is well known that branched-chain hydrocarbons tend in general to be more stable than less branched ones. In determining the parameters here, we need three values, which we have chosen to be methyl, iso, and neo. Methane is considered a special case. The methylene value is not included, which is equivalent to choosing it as our zero point by default. The neo value is most negative (approximately -6.9 kcal/mol), followed by the iso (-3.3 kcal/mol), the methylene (zero), and the methyl value, which is positive (-b2.0kcal/mol). [Pg.268]


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Equivalent position, special

Equivalent positions

Equivalent positions, general

Equivalents specials

General and special positions

General position

Position special

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