Group diperiodic

The combined symmetry of the top few planes determines the symmetry of the pattern. Because of this, attention has been directed to the diperiodic groups in two dimensions which are 80 in number (142) rather than to the 17 strictly plane groups (128) which do not admit symmetry operations involving the third dimension. At the present unsatisfactory stage of understanding of LEED intensities, application of these diperiodic groups in pattern interpretation is risky and is not recommended. 

Diperiodic Space Groups and Slab Models of Surfaces. 459... 

A semi-infinite crystal is a three-dimensional (3D) object. Its symmetry group contains, in addition to translations in the surface plane, only the rotations and mirror reflections that keep the atoms in the planes parallel to the surface. Only 17 such groups exist. Formally, these groups are isomorphic with diperiodic space groups in two dimensions. They are called plane groups. 

In the molecular-cluster approach a crystal with a surface is modeled by a finite system consisting of the atoms on the surface and of some atomic planes nearest to it. The diperiodicity of the surface is not taken into account. The symmetry of such a model is described by one of the crystallographic point groups. 

Diperiodic group DG contains a subgroup of two-dimensional translations with elements E a ), where... 

The symmetry of the single slab corresponds to one of 80 diperiodic (layer) space groups. Fig. 11.5 shows a 3-layer single-slab model of (001) surface of MgO crystal (each layer consists of one atomic plane). The symmetry group of this slab DG61 PAjmmm) belongs to a square system. 

Related to the spectroscopy of crystals is the spectroscopy of surfaces and, particularly, the spectroscopy of species adsorbed on crystal surfaces. For perfectly conducting metals, there is an important selection rule in that such surfaces image any electric dipole within an adsorbed molecule. When such dipoles are perpendicular to the surface the dipoles reinforce when they are parallel to the surface they cancel. This gives rise to the so-called surface selection rule , that it is only possible to observe by electric dipole spectroscopy those modes which involve dipole moment changes perpendicular to the surface. This requirement can be expressed group theoretically by use of the so-called diperiodic groups in two dimensions. 

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