2D Crystallography Selvage (or selvedge (it. cimosa)): Region in the solid in the vicinity of the mathematical surface Surface = Substrate (3D periodicity) + Selvage (few atomic layers with 2D periodicity) Warning: There may be cases where neither long-range nor short- range periodicity are given TLR-model Terrace-Ledge-Kink

2D Crystallography Bravais lattices in 2D are called Bravais nets Unit cells in 2D are called unit meshes There are just 5 symmetrically different Bravais nets in 2D The centered rectangular net is the only non-primitive net

2D Bravais Nets and Unit Meshes Rectangular (c) net |a 1 |≠|a 2 |  =90° Oblique (p) net |a 1 |≠|a 2 |  ≠90° Rectangular (p) net |a 1 |≠|a 2 |  =90° Square (p) net |a 1 |=|a 2 |  =90° a1a1 a2a2 a1a1 a2a2 a1a1 a2a2    Hexagonal (p) net |a 1 |=|a 2 |  =120° a1a1 a2a2 a2’a2’ a1’a1’ a1a1 a2a2    Primitive cell Unit cell |a 1 ’|≠|a 2 ’|  ≠90°

2D Crystallography: 2D Point Groups

2D Crystallography: 2D Space Groups The combination of the 5 Bravais nets with the 10 different point groups leads to 17 space groups in 2D (i.e. 17 surface structures) Equivalent positions, symmetry operations and long and short “International” notations for the 2D space groups

2D Crystallography: Relation between Substrate and Selvage Whenever there is a selvage (clean surface or adsorbate) the surface 2D-net and 2D-mesh are referred to the substrate 2D- net and 2D-mesh The vectors c 1 and c 2 of the surface mesh may be expressed in terms of the reference net a 1 and a 2 by a matrix operation (P) Since the area of the 2D substrate unit mesh is |a 1 xa 2 |, det G is the ratio of the areas of the two meshes

2D Crystallography: Relation between Substrate and Selvage Based on the values of det G and G ij, systems are sorted out along the following classification: 1) det G integral and all G ij integral The two meshes are simply related with the adsorbate mesh having the same translational symmetry as the whole surface 2) det G a rational fraction (or det G integral and some G ij rational) The two meshes are rationally related The structure is still commensurate but the true surface mesh is larger than either the substrate or adsorbate mesh. Such structures are referred to as coincidence net structures Now, if d 1 and d 2 are the primitive vectors of the true surface mesh, we have

2D Crystallography: Relation between Substrate and Selvage 2 continued) det G a rational fraction (or det G integral and some G ij rational) det P and det Q are chosen to have the smallest possible integral values and they are related by 3) det G irrational The two meshes are now incommensurate and no true surface mesh exists. This might be the case if the adsorbate-adsorbate bonding is much stronger than the adsorbate-substrate bonding or if the adsorbed species are too large and they do not “feel” the periodicity of the substrate

2D Crystallography: Relation between Substrate and Selvage Shorthand notation (E. A. Wood, 1964) It defines the ratio of the lengths of the surface and substrate meshes along with the angle through which one mesh must be rotated to align the two pairs of primitive translation vectors. If A is the adsorbate, X the substrate material and if |c 1 |=p|a 1 | and |a 2 |=q|c 2 | with a unit mesh rotation of , the structure is referred to as X{hkl}p x q-R  °-A or often X{hkl}(p x q)R  °-A Warning: This notation is less versatile. It is suitable for systems where the surface and substrate meshes have the same Bravais net, or where one is rectangular and the other square. It is not satisfactory for mixed symmetry meshes.

2D Crystallography: Surface Reciprocal Lattice The reciprocal net vectors c 1 * and c 2 * of the surface mesh are defined as c 1 c 2 * = c 2 c 1 * = 0 c 1 c 1 * = c 2 c 2 * = 2π (or 1) The reciprocal net points of a diperiodic net may be thought of (in 3D space) as rods. The rods are infinite in extent and normal to the surface plane where they pass through the reciprocal net points. Imagine a triperiodic lattice which is expanded with no limit along one axis, thus the lattice points along this axis are moved altogether and in the limit form a rod.

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