**1)** **Domains and symmetry:** A superlattice
frequently has lower symmetry than the basic 2D lattice. E.g. a (2x1)
superlattice on an ideal fcc(100) surface cannot have a 4-fold axis of
rotation, even though the fcc(100) does have a 4-fold axis of rotation.
However, the 4-fold axis of the basic lattice allows 4-fold rotated
"domains" of the (2x1) superlattice to coexist on different parts of
the surface. Each such symmetry-equivalent domain contributes a rotated copy of
the same diffraction pattern to the overall pattern, and, if these domains have
equally large areas, the resulting pattern will again have 4-fold symmetry. The
general rule is that symmetry-equivalent domains can together produce a LEED
pattern that has the same symmetry as that of the original basic 2D lattice. A
consequence of this is: a high-symmetry LEED pattern does NOT imply a lattice
(and thus structure) of equally high symmetry. In other words, a high-symmetry
LEED pattern is perfectly consistent with a low-symmetry single-domain
structure. (Note: the above discussion does not address spot intensities, but
spot locations in the pattern; at off-normal beam incidence, the spot intensities
will usually show no symmetry, while the spot locations still can form a
high-symmetry pattern.)

**2) Lattice geometry vs. atomic
structure:** LEEDpat
does not relate the space group to specific atomic positions (like adatom top,
bridge or hollow sites), since atomic positions are not defined in LEEDpat: it
is the user's responsibility to make any such connections with specific
structural models of a surface. The program SARCH does allow such connections
to be made, since it calculates spot intensities for predefined atomic
positions. However, it only uses kinematic theory, which is not reliable for
proper structure determination. Detailed surface structure determination by
LEED must take multiple scattering into account; corresponding computer codes
are available at http://www.icts.hkbu.edu.hk/surfstructinfo/SurfStrucInfo_files/leed/leedpack.html.

**3) Compatibility of symmetries
between basic 2D lattice and superlattice:** For a given space group of the basic 2D lattice, some
space groups are not allowed for the superlattice (e.g. a square substrate
lattice is incompatible with a hexagonal superlattice). LEEDpat analyses all
compatibility constraints and allows the user to only select compatible space
groups of the superlattice, namely those indicated above the input box for the
superlattice space group. This compatibility requirement corresponds to the
substrate "imprinting" its symmetries onto the surface structure in
the supercell. The analysis uses the following criteria to test compatibility:

a) The supercell and (1x1)
space groups are compatible if all symmetry elements of the superlattice are
contained in the group of the (1x1) lattice, i. e. all rotational axes, mirror
planes, and glide planes of the superlattice coincide with those of the (1x1)
lattice.

b) The 4-fold axes of the
(1x1) cell can also act as 2-fold axes, and the 6-fold axis of the (1x1) cell
can act as 2- and 3-fold axes of the supercell.

c) A glide plane of the
supercell can coincide with a mirror plane of the (1x1) cell if the repeat
length along the glide plane is a multiple 2n (= even number) of the repeat
length along the mirror plane. Therefore, on [3] pm a p(1x2) supercell cannot
exist as [4] pg whereas a p(2x1) can.

d) A glide plane of the supercell can coincide with a glide plane of the (1x1) cell if the repeat lengths along the planes are equal or differ by a multiple 2n+1 (= odd number). Therefore, on [4] pg the supercells p(1x1), p(3x1), p(5x1), ... can exist as [4] pg, whereas p(2x1), p(4x1), ... cannot.

For centered rectangular supercells the symmetry compatibility analysis relies
on basis vectors of equal length. Bases with vectors of different length can be
transformed accordingly by pressing the "Reduce" button.

The user can switch off/on compatibility checking by clicking at the checkbox
before "constrained" in the middle left panel (if the box is filled
with "v" checking is on, if the box is empty checking is off).

**4) Redundancy in matrix
notation, and Minkowski reduction (commensurate overlayers):** A given commensurate superlattice can
be represented by more than one matrix. E.g., a (2x2) superlattice on fcc(100)
can be represented by the matrix (2,0|0,2), which simply contains the two basis
vectors (2,0) and (0,2) of the superlattice relative to those of the basic 2D
lattice; but it can also be represented by the matrix (2,0|2,2), since the two
corresponding vectors (2,0) and (2,2) span the same superlattice. However, the
choice (2,0|0,2) is clearly more convenient and "natural". Any input
matrix can be "reduced" to a simpler and more conventional choice by
pressing the "Reduce" button. This executes a Minkowski reduction of
the superlattice basis vectors to obtain vectors of minimum length and included
angle closest to 90 degrees. It also selects a standard left-rotating pair of
vectors which is important for the compatibility analysis, see 3). For centered
rectangular lattices the reduction transforms to smallest vectors of equal
length.

**5) Incommensurate periodic
overlayers:** These
overlayers can be defined by 2x2 matrices (a11, a12 | a21, a22) with real
valued elements connecting the superlattice vectors (Rs1, Rs2) with those of
the basic lattice (R1, R2) by

Rs1 = a11 R1 + a12 R2, Rs2 = a21 R1 + a22 R2

While both the basic 2D and the superlattice are periodic, the combination of the two has no periodicity. However, if independent scattering of the two lattices is assumed this results in a superposition of the corresponding LEED patterns. Assuming further that the basic 2D lattice imposes its symmetry on the superlattice orientation, resulting in different domains (see above), the resulting LEEDpattern can be represented by a superposition of the pattern of each domain orientation. Lattice vectors of incommensurate overlayers can also be Minkowski reduced analogous to the commensurate case, see 4).

Special cases of incommensurate periodic overlayers are those where the superlattice vectors (Rs1, Rs2) arise from those of a commensurate structure by (small) rotations or scaling. LEEDpat includes options to define such overlayers. Resulting Moiré patterns in real space may become apparent by inspecting larger surface sections.

**6) Hexagonal substrate and
overlayers:** Substrate
and overlayer lattice vectors referring to hexagonal lattices can be described
equivalently using obtuse or acute representations, where affect corresponding
superlattice matrices. LEEDpat allows both representations and converts superlattice
matrices accordingly.

**7) Higher-order coincidence (HOC)
lattices: **here the
basic transformation matrix (M) connecting overlayer and substrate periodicity
contains only fractional (rational) values pij/qij where pij, qij are both
integers. This can be written also as a product of an inverted matrix of (B)
and a matrix (A), i.e. (M) = (B)^-1 (A). Here (A), (B) are integer valued and
characterize lattice vectors of the common coincidence supercell (substrate and
overlayer combined) in the basis of the substrate and overlayer, respectively.

As an illustration, the usual (n x m) superlattice situation over a (1x1)
substrate, with e.g. one adsorbate at each superlattice grid point, corresponds
to (B) equal to the unit matrix (1) and (A) = (M) describing the matrix of the
(n x m) superlattice. A low-order example is a (2x1) superlattice, where one
lattice constant is doubled relative to (1x1). A higher-order coincidence (HOC)
lattice arises when an overlayer has a fractional relationship with the
substrate, such as (3/2 x 1), where adsorbates at its grid points will
alternately fall on vs. between the substrate grid points. This superlattice is
then more logically described as having a (3x1) periodicity relative to the substrate
lattice and a (2x1) periodicity relative to the overlayer lattice, i.e. we need
to move forward by 3 substrate cells, or equivalently by 2 overlayer cells, to
find coincidence between the two lattices. In this example, matrix (A)
corresponds to (3x1) while matrix (B) corresponds to (2x1). Higher-order
coincidence lattices generalize this concept allowing any combination of
integer valued matrices (A), (B) where the resulting transformation matrix is
fractional according to (M) = (B)^-1 (A).

LEEDpat allows separately changing fractional (M) as well as integer (A) and
(B) and converts automatically between these matrices. It also evaluates
coincidence supercell lengths and sizes.