2D Crystallography

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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)
TLR-model Terrace-Ledge-Kink
Warning There may be cases where neither
long-range nor short-range periodicity are given
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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
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2D Crystallography 2D Point Groups
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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
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Equivalent positions, symmetry operations and
long and short International notations for the
2D space groups
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Equivalent positions, symmetry operations and
long and short International notations for the
2D space groups
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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 c1 and c2 of the surface
mesh may be expressed in terms of the reference
net a1 and a2 by a matrix operation (P)
Since the area of the 2D substrate unit mesh is
a1xa2, det G is the ratio of the areas of the
two meshes
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2D Crystallography Relation between Substrate
and Selvage Based on the values of det G and Gij,
systems are sorted out along the following
classification 1) det G integral and all Gij
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 Gij
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
d1 and d2 are the primitive vectors of the true
surface mesh, we have
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2D Crystallography Relation between Substrate
and Selvage 2 continued) det G a rational
fraction (or det G integral and some Gij
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
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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 c1pa1 and a2qc2 with a
unit mesh rotation of f, the structure is
referred to as Xhklp x q-R f-A or often
Xhkl(p x q)R f-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.
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2D Crystallography Surface Reciprocal
Lattice The reciprocal net vectors c1 and c2 of
the surface mesh are defined as c1 c2 c2
c1 0 c1 c1 c2 c2 2p (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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Ricostruzioni e superreticoli
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