Coordinate fractional

Band structure calculations have been done for very complicated systems however, most of software is not yet automated enough or sufficiently fast that anyone performs band structures casually. Setting up the input for a band structure calculation can be more complex than for most molecular programs. The molecular geometry is usually input in fractional coordinates. The unit cell lattice vectors and crystallographic angles must also be provided. It may be nee-... 

The program uses two ASCII input files for the SCF and properties stages of the calculation. There is a text output file as well as a number of binary or ASCII data files that can be created. The geometry is entered in fractional coordinates for periodic dimensions and Cartesian coordinates for nonperiodic dimensions. The user must specify the symmetry of the system. The input geometry must be oriented according to the symmetry axes and only the symmetry-unique atoms are listed. Some aspects of the input are cumbersome, such as the basis set specification. However, the input format is documented in detail. 

X thermal exergy (i.e., exergy associated with heat Q) y mole fraction coordinate... 

The fractional coordinates in crystallography, referred to direct and reciprocal cells respectively, are examples of covariant and contravariant vector components. 

Nj is the coordination number, equal to the number of neighbors in the fh coordination shell. For an fee metal such as rhodium we expect 12 neighbors in the first shell. If a particle becomes small, the average coordination number decreases. Note that unless the sample is that of a single element, TV is a fractional coordination number, i.e. the product of the real coordination number and the concentration of the... 

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). 

The unit cell content. To complete the description of the crystal structure, the list of the atoms contained in the unit cell and their coordinates (fractional coordinates related to the adopted system and unit cell edges) are then reported. These are usually presented in a format such as M El in n x, y, z. In the MoSi2 structure, also reported in Table 3.2, and in Fig. 3.7, for instance, four silicon atoms... 

Figure 3.8. Crystal structure of CsCl. The positions of the centres of the atoms in the unit cell are shown in (a). In (b) the same cell is described by means of its characteristic sections taken at the height 0, A, and 1 of the third axis. In (c) a projection of the cell on its square basis is presented the values of the third (fractional) coordinate are indicated. In (d) the shortest interatomic distances are shown dCs-ci = a)3/2 = 411.3 X 0.866025. = 356.2. In (e) the subsequent group of interatomic distances (d = a = 411.3) involving six atoms in the adjacent cells is presented. A group of eight cells is represented in (f) to suggest that the actual structure of CsCl corresponds to a three-dimensional infinite repetition of unit cells and to show that the coordination around the white atoms is similar to that around the black ones shown in (d). The unit cell of the CsCl structure is shown as a packed spheres model in (g).

The coordinates indicated in the reported partial list of invariant lattice complexes correspond to the so-called standard setting and to related standard representations. Some of the non-standard settings of an invariant lattice complex may be described by a shifting vector, defined in terms of fractional coordinates, in front of the symbol. The most common shifting vectors also have abbreviated symbols P represents 14, A,AP (that is the coordinates which are obtained by adding A, Vi, Ai to those of P, that is coordinates 14, 14, A), J represents A, A, A J (coordinates A, 0, 0 0, A, 0 0, 0, A) F" represents A,A,AF (coordinates At, A, A A, 3A, A 3A, A, 3A 3A, A, A) and F" represents A, /, 3A F. It can be seen, moreover, that the complex D corresponds to the coordinates F + F". 

WoWj/2 the body-centred cubic structure of W (1 atom in 0, 0, 0 and 1 atom in A, A, /) corresponds to a sequence of type 1 and type 4 square nets at the heights 0 and A, respectively. Note, however, that for a fall description of the structure, either in the hexagonal or the tetragonal case, the inter-layer distance must be taken into account not only in terms of the fractional coordinates (that is, the c/a axial ratio must be considered). For more complex polygonal nets, their symbolic representation and use in the description, for instance, of the Frank-Kasper phases, see Frank and Kasper (1958) and Pearson (1972). 

Herein denotes fj the atomic scattering factor of atom j in the unit cell, the Xy jy Zj are the corresponding fractional coordinates of the atom j and the hkl are the Miller) indices of the Fourier component (see below). If the structure is... 

Because we can always choose each atom so it lies within the supercell, 0 < j), < 1 for all i and j. These coefficients are called the fractional coordinates of the atoms in the supercell. The fractional coordinates are often written in terms of a vector for each distinct atom. In the hep structure defined above, for example, the two atoms lie at fractional coordinates (0,0,0) and (5,5, 5). Notice that with this definition the only place that the lattice parameters appear in the definition of the supercell is in the lattice vectors. The definition of a supercell with a set of lattice vectors and a set of fractional coordinates is by far the most convenient way to describe an arbitrary supercell, and it is the notation we will use throughout the remainder of this book. Most, if not all, popular DFT packages allow or require you to define supercells using this notation. 

The position of an atom or ion in a unit cell is described by its fractional coordinates these are simply the coordinates based on the unit cell axes (known as the crystallographic axes), but expressed as fractions of the unit cell lengths. It has the simplicity of a universal system which enables unit cell positions to be compared from structure to structure regardless of variation in unit cell size. 

FIGURE 1.50 (a) The crystal structure of CO2, (b) packing diagram of the unit cell of CO2 projected on to the xy plane. The heights of the atoms are expressed as fractional coordinates of c. C, blue spheres 0, grey spheres. 

The crystal structure of the 1-2-3 superconductor, YBazCusOy- is depicted in Figure 10.8. Figure 10.8(a) depicts only the positions of the metal atoms. If we discuss it in terms of the perovskite structure ABO3, where B=Cu, the central section is now an A-type perovskite unit cell and above and below it are also A-type perovskite unit cells with their bottom and top layers missing. This gives copper atoms at the unit cell corners and on the unit cell edges at fractional coordinates A and Ys. The atom at the body-centre of the cell (i.e., in the centre of the middle section) is yttrium. The atoms in the centres of the top and bottom cubes are barium... 

Fractional Coordinates. In specifying the location of a point in a crystal lattice it is customary to employ coordinates, jt, y, z, that give the fraction of each principal vector distance (a, b, c), which define the unit cell. Thus, a point at the origin has the fractional coordinates 0,0,0 while the center of the cell has the coordinates 3,2,5. The face centers are 0, , , ,0, and , ,0 for the a, b, and c faces, respectively. It is to be emphasized that these fractional coordinates are not Cartesian except for isometric cells and are not even orthogonal for triclinic, monoclinic, or hexagonal lattices. 

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. 

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