Coordination equivalent fraction

A perspective view of the graphical symbols used in the International Tables (Hahn 2002) for the different axes is shown, (t is the shortest translation vector in the direction of the axes). The projection (along these axes) on the base plane of the equivalent points is also shown notice that the same projection is obtained in all the cases illustrated. The coordinates of all the equivalent points in the different sets are listed. Notice that the x, y, z coordinates are fractional coordinates they indicate the positions along the corresponding directions as fractions of the constants a, b and c (in these examples c = t). 

Again, we switch from d3r - dxdydz to the equivalent fractional coordinates Vd3a Vdfdijdf so as to circumvent the problem of variable limits. Then... 

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. 

Table 16. Fractional atomic coordinates, equivalent temperature factors and final anisotropic chemical parameters (with e.s.d. s in parentheses) for compound (III)...

Note that subtracting an amount log a from the coordinate values along the abscissa is equivalent to dividing each of the t s by the appropriate a-p value. This means that times are represented by the reduced variable t/a in which t is expressed as a multiple or fraction of a-p which is called the shift factor. The temperature at which the master curve is constructed is an arbitrary choice, although the glass transition temperature is widely used. When some value other than Tg is used as a reference temperature, we shall designate it by the symbol To. 

These observations are equivalent to a coarse-grained view of the system, which is tantamount to a description in terms of continuum mechanics. [It is clear that "points" of the continuum may not refer to such small collections of atoms that thermal fluctuations of the coordinates of their centers of mass become substantial fractions of their strain displacements.] The elastomer is thus considered to consist of a large number of quasi-finite elements, which interact with one another through dividing surfaces. 

Figure 5.19 shows the evolution of the gel volume fraction as a function of reduced time K(t — to). The solid line represents the theoretical curve obtained using Eq. (5.22). For (p values between 20% and 80%, the theoretical curve is roughly linear with a slope of 1 (see dashed line). Equivalently, within the same (p range, the volume fraction should vary linearly with time with a slope equal to K. The experimental data (Fig. 5.17) were recalculated in order to be plotted in reduced coordinates. For each initial volume fraction, K is deduced from the initial slope of the curve (p = f t) (for cp between roughly 20% and 60%). All the data lie within a unique curve that is in reasonable agreement with the theoretical one. 

For example, the center atom in the BCC space lattice (see Figure 1.20) has cell coordinates of 1/2, 1/2, 1/2. Any two points are equivalent if the fractional portions of their coordinates are equal ... 

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. 

Correlation diminishes the effectiveness of atomic jumps in diffusional random motion. For example, when an atom has just moved through site exchange with a vacancy, the probability of reversing this jump is much higher than that of making a further vacancy exchange step in one of the other possible jump directions. Indeed, if z is the coordination number of equivalent atoms in the lattice, the fraction of ineffective jumps is approximately 2/z (for sufficiently diluted vacancies as carriers) [C. A. Sholl (1992)]. 

As was already mentioned, due to the Pauli exclusion principle, which states that no two electrons can have the same wave functions, a wave function of an atom must be antisymmetric upon interchange of any two electron coordinates. For a shell of equivalent electrons this requirement is satisfied with the help of the usual coefficients of fractional parentage. However, for non-equivalent electrons the antisymmetrization procedure is different. If we have N non-equivalent electrons, then a wave function that is antisymmetric upon interchange of any two electron coordinates can be formed by taking the following linear combination of products of one-electron functions [16] ... 

Number m is the maximum, spatially allowable, number of B s in a coordination sphere surrounding A. Therefore, the answer to the above question is equivalent to a characterization of W (P) which we shall show to be equal to (i n 1.a where a is the fractional atomic composition of A. 

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. 

The relationship between cubic close-packed (ccp) structures and ionic compounds of type B1 is obvious. Interstitial sites with respect to metal positions are at fractional coordinates of the type 00 and equivalent to the ionic sites in Bl. The Madelung constant of Al type metals with interstitially localized free electrons is therefore the same as that of rocksalt structures. It is noted that the interstitial sites define the same face-centred lattice as the metal ions. 

To explore the periodic structure of the set Sk, and hence of the stable nuclides, it is convenient to represent each fraction h/k by its equivalent Ford circle of radius rp = 1/2k2, centred at coordinates h/k, rp. Any unimodular pair of Ford circles are tangent to each other and to the x-axis. If the x-axis is identified with atomic numbers, touching spheres are interpreted to represent the geometric distribution of electrons in contiguous concentric shells. The predicted shell structure of 2k2 electrons per shell is 2, 8, 8, 18, 18, 32, 32, etc., with sub-shells defined by embedded circles, as 8=2+6,... 

Translational symmetry is assumed in both directions, X and Y. This assumption implies that if an operator moves any site into a neighbouring cell it is equivalent to that site entering the reference cell from the opposite side. The coordinates x,y are therefore fractional and symmetry translations are unity, whereby 1 + x x x — 1 — x, etc. 

Figure 2.12. Schematic of the packing in an hep unit cell, showing equivalent close-packed B sites. The hexagon of A atoms is referred to as the basal plane, as this plane marks the top and bottom of a unit cell. The fractional coordinates of each set of B sites are (a, b,c) (1/3,2/3,1/2) - shown, or (2/3,1/3,1/2). ...

As briefly mentioned in the previous section, equivalent positions (or sites) that are listed in the field No. 8 in Table 1.18 for each crystallographic space group, represent sets of symmetrically equivalent points found in one unit cell. All equivalent points in one site are obtained from an initial point by applying all symmetry operations that are present in the unit cell. The fractional coordinates (coordinate triplet) of the initial (or independent) point are usually marked as x, y, z. 

We know that any symmetry operator, such as a twofold axis or a sixfold screw axis, is equivalent to a set of general fractional coordinates in the unit cell where, by symmetry, corresponding atoms on different asymmetric units are found (see Chapter 3). Thus to specify that there is a twofold axis along y (and it is always essential to specify the direction of the operator) is the same as saying that for an atom at any x, y, z, there must be an identical atom at —x, y, —z. To say there is a axis along z is equivalent to stating... 

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