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of longer range - mediated by substrate electrons or the elastic field. Elastic inter-
actions tend to be of one sign and decay monotonically with adsorbate separation
distance
d
, like
d
−
3
asymptotically. They are generally taken to be isotropic, even
when unjustified by the elastic tensor, since computing elastic Green's functions in
the anisotropic case is notoriously difficult [
28
]. The electronic indirect interaction
has richer behavior, oscillating in sign and reflecting the isotropy or anisotropy of
the substrate wavefunctions in the surface plane [
29
]. Asymptotically it is dominated
by the Fermi wavevector and has the Friedel behavior
d
−
n
sin
(
2
k
F
d
+
)
(2.2)
where
d
is the [lateral] distance between the adsorbates;
this result
is non-
perturbational, the phase factor
distinguishing it from the well-known perturba-
tional RKKY expression [
30
]. (For a non-isotropic Fermi surface, the appropriate
wavevector has
velocity
parallel to
d
;see[
3
] for details.) In the bulk,
n
3, but at
a metal surface the leading term is canceled by that from its image charge, yielding
n
=
5. For trio interactions an expression similar to (
2.2
) holds to lowest order,
with
d
replaced by the perimeter of the triangle formed by the adatoms. Here the
decay is even faster, with
n
=
7. In short, interactions mediated by bulk states have
negligible strength for values of
d
for which the asymptotic expression is valid.
The situation is strikingly different when there is a metallic surface state (i.e.,
a surface band that crosses the Fermi energy), such as found near the
=
¯
point on
the (111) faces of noble metals (associated with the (111) necks of the Fermi sur-
face). For this case
n
is 2 and 5/2 for pair and trio interactions, respectively, so
that the asymptotic regime is physically important [
31
]. Indeed, trio interactions
may play a role in the formation of 2D clusters of Cu/Cu(110) [
32
]. Furthermore,
the Fermi “surface” is circular, and
k
F
is much smaller than its bulk counterpart,
leading to the dramatic oscillations (with wavelength
π/
k
F
≈
15 Å) seen in STM
experiments [
33
].
Most of the content of this and the second, third, and fifth sections of this chapter
were included in the first author's presentation at the Vth Stranski-Kaischew Surface
Science Workshop (SK-SSW'2005): “Nanophenomena at Surfaces - Fundamentals
of Exotic Condensed Matter Properties” in Pamporovo, but have been updated. Each
section provides references from which the content was adapted and from which
more information can be obtained.
2.2 Recollection of Two Effects on Statistical Mechanics
In this section we recall two remarkable implications of trio interactions for phase
diagrams [
15
]. While these ideas are not new, they have been largely ignored by the
community and so bear repeating.
