Unit cell atomic coordinates

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. 

In this section, we concentrate on the relationship between diffraction pattern and surface lattice [5], In direct analogy with the tln-ee-dimensional bulk case, the surface lattice is defined by two vectors a and b parallel to the surface (defined already above), subtended by an angle y a and b together specify one unit cell, as illustrated in figure B1.21.4. Withm that unit cell atoms are arranged according to a basis, which is the list of atomic coordinates within drat unit cell we need not know these positions for the purposes of this discussion. Note that this unit cell can be viewed as being infinitely deep in the third dimension (perpendicular to the surface), so as to include all atoms below the surface to arbitrary depth. 

The minimum amount of information needed to specify a crystal structure is the unit cell type, that is, cubic, tetragonal, and so on, the unit cell parameters, and the positions of all of the atoms in the unit cell. The atomic contents of the unit cell are a simple multiple, Z, of the composition of the material. The value of Z is equal to the number of formula units of the solid in the unit cell. Atom positions are expressed in terms of three coordinates, x, y, and z. These are taken as fractions of a, b, and c, the unit cell sides, say and j. The x, y, and z coordinates are plotted with respect to the unit cell axes, not to a Cartesian set of axes. The space group describes the symmetry of the unit cell, and although it is not mandatory when specifying a structure, its use considerably shortens the list of atomic positions that must be specified in order to built the structure. 

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. 

Presented band structure calculations have been done by computer code SOLID 2000 [4,5], The lattice parameters of MgB2 (hexagonal structure, space group P6mmm), with the fraction coordinates of the unit cell atoms Mg = (0,0,0) B1 = (1/3,2/3,1/2) B2 = (2/3,1/3,1/2), have been optimized in a good agreement with the experiment, a/ Equilibrium - undistorted geometry... 

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). 

Translational symmetry requires that any matter located at xea + yeb + zec must also be replicated exactly at the coordinates (x + l a ) ea + (y + m b ) eb + z + n c ) ecr where /, m, and n are integers (positive, negative, or zero). This translation symmetry defines the unit cell and the unit cell axes a, b, c. In the least symmetric case, the contents of the unit cell (atoms, ions, molecules, trapped solvents, proteins) may not have any symmetry at all. 

A positron is defined as a set of symmetrically equivalent coordinate points. Within the unit cell, atoms or molecules may be located at general positions that do not lie on any symmetry element or at special positions. If they do lie either on a symmetry... 

Like the diamond stracmre discussed earlier, the honeycomb stracture is not itself a Bravais lattice. If the lattice is translated by one nearest-neighbor distance, the lattice does not go into itself. There are two nonequivalent, or distinct types of sites per unit cell, atoms a and b, separated by a distance Uq, as shown later in Figure 4.6. However, a Bravais lattice can be created by taking this pair of distinct atoms to serve as the basis. Doing so, shows that the vectors of the two-dimensional hexagonal lattice, a and U2, are primitive translation vectors. A given site on one sublattice with coordinates (0, 0), has three nearest neighbors on the other sublattice. They are located at (0, U2), (fli, 0), and (- , 0). 

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.

An atom at a cell comer is given the coordinates (0, 0, 0). An atom at the centre of the face of a unit cell is given the coordinates ( A, A, 0) if it lies between the a- and b-axes, ( A, 0, A ) if between the a- and c-axes, and (0, A, V2) if between the b- and c-axes. An atom at the centre of a unit cell would have a position specified as (Z2, V2, V2), irrespective of the type of unit cell. Atoms at the centres of the cell edges are specified at positions (V2, 0, 0), (0, V2, 0) or (0, 0, A), for atoms on the a-, b- and c-axis, (Figure 1.6). Stacking of the unit cells to build a structure will ensure that an atom at the unit cell origin will appear at every comer, and atoms on unit cell... 

The summation in Eq. 15 is over all atoms of the unit cell with coordinates xt, yu and Z. fi is an atomic scattering factor. 

After constructing the graph G = (V, E) for the structure under study we can construct its adjacency matrix A and its Laplacian Q as well. From the Eqs. 6.10-6.21 follows that we do not need the special Descartes coordinates of unit cell atoms for the determination of the connectivity graph G = (V,E). The initial data contain only the integers m, n, p, q and the list of neighbours for each unit cell atoms. A neighbouring atom is described by two numbers the serial number r and the type number t. 

Due to the projection of the electronic density investigated in the plane (x, y) of the unit cell, the coordinates of the atoms of Si, r, that contribute to the electronic density will be also limited to the set ... 

Fig. 3. Projection of the Pr] j,Co9Ge4 unit cell and coordination polyhedra of the atoms (a) Pr, (b-e) Co, Ge, 3.2. R(M,Ge)i2 compounds 3.2.1. YCogGeg structure type...

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