Plane groups and invariant functions

For most of the surfaces of interest, in addition to the two-dimensional translational symmetry, there are additional symmetry operations that leave the lattice invariant. If the tip has axial symmetry, then the STM images and the AFM images should exhibit the same symmetry as that of the surface. The existence of those symmetry elements may greatly reduce the number of independent parameters required to describe the images. 

The collection of all symmetry operations that leave a crystalline lattice invariant forms a space group. Each type of crystal lattice has its specific space group. The problem of enumerating and describing all possible space groups, both two dimensional three dimensional, is a pure mathematical problem. It was completely resolved in the mid-nineteenth century. A contemporary tabulation of the properties of all space groups can be found in Hahn (1987). Bums and Glazer (1990) wrote an introductory book to that colossal table. 

Compared with the problem in three-dimensional space, which has 14 Bravais lattices and 230 space groups, the problem of surface symmetry is tmly a dwarf It only has 5 Bravais lattices and 17 different groups. The five Bravais lattices are listed in Table E.l. 

As shown in Table E.l, there is only one centered lattice, oc. It is easy to show that for monoclinic, orthorhombic, and hexagonal cases, the centered lattice reduces to primitive lattices with halved unit cells. 

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