Cell, unit coordinates

Figure 3.23. Hexagonal, 63, net of points, a, b and c are the codes corresponding to the different positions relative to the cell origin. Notice that the unit cell contains two points (every point in a comer is in common with four adjacent hexagonal cells). The coordinate doublets of the points are reported.

Figure 3.25. Stacking symbols applicable to larger unit cells. A, B, C three nodes in the unit cell of a triangular net a, b, c six nodes of a hexagonal net in the unit cell a, 3, p nine nodes of a kagome net in the unit cell. The coordinate doublets are indicated.

From crystallography, we obtain an image of the electron clouds that surround the molecules in the average unit cell in the crystal. We hope this image will allow us to locate all atoms in the unit cell. The location of an atom is usually given by a set of three-dimensional Cartesian coordinates, x, y, and z. One of the vertices (a lattice point or any other convenient point) is used as the origin of the unit cell s coordinate system and is assigned the coordinates x = 0, y = 0, and z = 0, usually written (0,0,0). See Fig. 2.4. 

Fig. 21. Projection of the NdOs4Sbj2 unit cell and coordination polyhedra of atoms.

The crystal structure of many pure metals adopts the hep of identical spheres, which has a primitive hexagonal lattice in space group D h - Pbi/mmc. There are two atoms in the hexagonal unit cell with coordinates (0,0,0) and(f,i,i). 

It is frequently formd that it is not possible to find a primitive rrrrit cell with edges parallel to crystal axes chosen on the basis of symmetry. In such a case the crystal axes, chosen on the basis of symmetry, are proportiorral to the edges of a unit of stracture that is larger than a primitive unit cell. Such a rrrrit is called a nonprimitive unit cell, and there is more than one lattice point per nonprimitive unit cell. If the nonprimitive unit cell is chosen as small as possible consistent with the symmetry desired, it is found that the extra lattice points (those other than the comer points) lie in the center of the unit cell or at the centers of some or all of the faces of the imit cell. The coordinates of the lattice points in such a case are therefore either integers or half-integers. 

Other complexes of gallium have been more exhaustively studied. Gallium hydride, GaH3, forms with trimethylamine a 1 1 solid complex which melts at 69°C. and decomposes slowly even at room temperature (23), X-ray diffraction studies of the 1 1 complex show the presence of H3GaN(CH3)3 units with two molecules per unit cell. A coordination number of 4 for gallium is indicated (21). 

Atomic parameters A set of numbers that specifies the position of an atom in the unit cell (atomic coordinates), the extent of its displacement about an equilibrium position (vibration), and an occupancy factor (generally 1.0). Three parameters define position, one parameter can be used to define isotropic displacements, or six to define anisotropic displacements and one parameter defines the site occupancy. 

Fig. 7.2 Fluorite UOj structure contains eightfold coordinated U atoms (grey spheres) and fourfold coordinated O atoms (black spheres)-, the cell unit length a=b = c = 5.47A. Uranium cations can have an oxidation state greater than U +, as a result of the removal of two electrons from the 5/ orbital, to become The UO system can be used as a model for oxidation/reduction reactions since it is an intermediate between the uranium metal (U ) phase and the most oxidized uranium phase (UO )...

Figure 1.5. Illustration of the content of the unit cell. The coordinates of the center of each atom are given as doublets, i.e. Xi, Xi, y2 and Xf 73. In three dimensions, they become triplets, i.e. a ,-, yi, Zj.

Thus, in the expanded form, the value of the electron density at any point in the unit cell with coordinates x, y and z(0 calculated using structure amplitudes obtained from x-ray diffraction experiment as ... 

The energy Euc in Eq. (33) can be E JiPI(2> or other correlated total energy per unit cell. The coordinates Y A are Fourier transforms of the nuclei s Carte-... 

Figure 1. Laumontite Am, first setting, lower part of the unit cell, c-coordinates from 0.0 to 0.5...

By choosing two non-collinear translations a and b in Fig. 1.8(a), we describe the lattice by the translation vectors r = wa H- t b,M and v being integers. We call this coordinate system the lattice base. The parallelogram (a, b) is the cell (unit cell). Analogously, the base a, b, c of a three-dimensional lattice is defined by three non-coplanar translations. The cell is hence a parallelepiped. The coordinates x, y, z of a point inside this cell are referred to this non-unitary coordinate system. The set of all the points equivalent by translation to the point Xpyp Zj is given by... 

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