Direct Space Lattice Parameters

A crystal is a periodic array of ordered entities (e.g., ions, atoms, molecules) in three dimensions. The repeating unit is imagined to be a unit cell whose volume and shape are designated by the three vectors representing the length and direction of the cell edges as three unit vectors of translation. 

Rgure D.1. lUCr standardized notation for space lattice parameters 

A space lattice is defined by either the three unit lattice vectors a, b, and c or the set of the six lattice parameters a, b,c, a, fi, and where the last three quantities represent the plane angles between the cell edges. The International Union of Crystallography (lUCr) has now standardized the notation and definition of space lattice parameters and this international standard nomenclature is listed below  

There are seven possible space lattices which entirely describe both inorganic and organic crystalline materials. These are called the seven crystal systems (i.e., cubic, tetragonal, hexagonal, trigonal, orthorhombic, monoclinic, and triclinic). 

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There is only one space lattice in the rhombohedral crystal system. This crystal is sometimes called hexagonal R or trigonal R, so don t confnse it with the other two similarly-named crystal systems. The rhombohedral crystal has nniform lattice parameters in all directions and has equivalent interaxial angles, bnt the angles are nonorthogonal and are less than 120°. 

In direct analogy with two dimensions, we can define a primitive unit cell that when repeated by translations in space, generates a 3D space lattice. There are only 14 unique ways of connecting lattice points in three dimensions, which define unit cells (Bravais, 1850). These are the 14 three-dimensional Bravais lattices. The unit cells of the Bravais lattices may be described by six parameters three translation vectors (a, b, c) and three interaxial angle (a, (3, y). These six parameters differentiate the seven crystal systems triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. 

Lattice parameters of graphitelike carbon crystals along a e c directions. Interlayer spacing. 

A crystal structure is described by a collection of parameters that give the arrangement of the atoms, their motions and the probability that each atom occupies a given location. These parameters are the atomic fractional coordinates, atomic displacement or thermal parameters, and occupancy factors. A scale factor then relates the calculated structure factors to the observed values. This is the suite of parameters usually encountered in a single crystal structure refinement. In the case of a Rietveld refinement an additional set of parameters describes the powder diffraction profile via lattice parameters, profile parameters and background coefficients. Occasionally other parameters are used these describe preferred orientation or texture, absorption and other effects. These parameters may be directly related to other parameters via space group symmetry or by relations that are presumed to hold by the experimenter. These relations can be described in the refinement as constraints and as they relate the shifts, Ap,-, in the parameters, they can be represented by... 

Diffraction spots reveal the symmetry and the spacing of the reciprocal lattice. Since both of them are directly related to the crystal lattice, diffraction patterns recorded for different crystal orientations can be used to determine crystal symmetry and lattice parameters. In order to see sharp diffraction spots, the sample has to be illuminated by a plane wave. A plane-wave illumination warrants that only one incident wave vector ko goes into the elastic scattering relation, k ko = q. A plane-wave illumination, however, means that the entire sample is uniformly illuminated. The information contained in such a diffraction pattern is not localizable. 

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