Periodic Space Tiling and Crystal Structures

Cleavage of crystals, in particular of calcite CaC03, and the law of rational indices generated the idea of the periodicity of crystal structures and the theory of translational lattices  

A crystal structure consists of a periodic repeat in three dimensions of some motif 

This theory implies the existence of a microscopic unit of structure (the molecule integrante of Rene Just Hauy), which played as fundamental a role in the discovery of atoms as the laws of chemical stoichiometry. The X-ray diffraction experiment (M. von Laue, 1912 Chapter 3) provided brilliant confirmation of its existence. 

We consider that the two-dimensional periodic structure of Fig. 1.8(a) extends to infinity. Let us choose some dot and all the dots equivalent to this one. We call this set of dots in Fig. 1.8(a) the translational lattice, or simply the lattice. The translation of the diagram from one dot to another dot is an operation which yields an invariance, i.e, this is a symmetry operation. We call these dots lattice points. Other examples of two-dimensional periodic structures are given by patterned wallpapers. 

The periodically repeating unit is called a motif. In Fig. 1.8(a), the contents of one of the parallelograms can be considered to be the motif. It is important to distinguish clearly between the terms lattice, motif and structure  

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