Surface Lattice and Superstructure

In order to describe the positions of surface atoms, a 2D reference frame has to be introduced. Due to the 2D periodicity along the surface it is sufficient to specify the arrangement of atoms in a unit cell. Besides the translational invariance, a crystal possesses a symmetry with respect to rotations and reflections in mirror planes. At a crystal surface, these point-group operations have to leave the surface atoms in the surface plane. Therefore, only rotation axes and mirror planes perpendicular to the surface can be included in a surface symmetry group. The possible point-group operations must be compatible with 2D translations. This condition leads us to the following ten allowed point-group symmetries  

the digits n = 1...6 denote the axes of rotation, C , by an angle 2n/n and the letter m implies a mirror plane (Tv containing a rotation axis. 

Any crystal plane can be specified by a 2D Bravais lattice constructed of primitive (p) 2D unit cells (parallelograms) with a lattice point at each corner. The symmetry considerations given above restrict the number of different 

The areas of the primitive cells of the two Bravais lattices are equal to A = ai X Azl and B = bi x b2. As follows from Eq. (2.6), they are related to each other by the equation 

The value of det C thus characterizes the distortion of the ideal crystal structure due to cleavage. When det C is an integer, the superlattice is referred to as simple (Figs 2.2a, b). When detC is a rational number, the superstructure is called coincidence lattice. In such a case, if there is no angle between the 

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