Chemistry Reference
In-Depth Information
ordered phase compared to the higher temperature disordered phase. Evidently the
breaking by trios of the particle-hole symmetry of the pair interaction lattice-gas
hamiltonian, and the resulting asymmetry in the ground state energy, is not the crux.
We describe a crude approximation scheme for assessing the change in the dis-
ordering temperature
T
c
of an ordered phase from a known value
T
c
(
0
)
for some
Hamiltonian to
T
c
(
for a more complicated Hamiltonian with a new interac-
tion energy
E
new
. While we have applied our procedure [
36
] to a wide range of
problems, we still have no formal derivation. In essence, the idea is that
T
c
scales
with the lowest energy excitation from the ground state. In [
36
] we show, e.g., that
for a
c
E
new
)
overlayer with half-monolayer coverage, characterized by a nearest-
neighbor repulsion
E
1
and a smaller second-neighbor interaction
E
2
,
(
2
×
2
)
1
4
E
2
3
E
1
T
c
(
E
2
)
=
T
c
(
E
2
=
0
)
−
(2.6)
For this simple problem, Barber [
37
] showed that the exact coefficient is
√
2
≈
1.41 rather than 4/3
1.33; on the other hand, our value is much better than the
mean-field prediction of 1. For this same problem, the effect of a right-triangle trio
interaction
E
RT
(with
E
2
=
≈
0) is given by
1
3
E
1
+
2
E
RT
3
E
1
T
c
(
2
E
RT
3
E
1
=
⇒
T
c
(
E
RT
)
=
T
c
(
E
RT
=
0
)
+
(2.7)
T
c
(
E
RT
)
0
)
Similarly, for a linear trio
E
LT
1
3
E
1
+
3
E
1
T
c
(
E
LT
E
LT
3
E
1
=
⇒
T
c
(
E
LT
)
=
T
c
(
E
LT
=
)
+
0
(2.8)
T
c
(
E
LT
)
)
0
We caution that this procedure is applicable only if the new interaction does not
alter the symmetry of the ordered state and works well only if the nearby elementary
excitation from the fully ordered state is uniquely defined. Thus, it works well for a
(
√
3
×
√
3
. It is also curious
that this procedure requires a lattice-gas picture in which the number of atoms is
conserved (i.e., a canonical ensemble or Kawasaki dynamics); if the atom instead
hopped to a “bath” (i.e., a grand canonical ensemble, or fixed chemical potential, or
a single spin flip in a spin analogue (Glauber dynamics)), the predictions are quite
poor.
To assess the effect of trios on the symmetry of the temperature-coverage phase
boundary of a
c
)
overlayer on a hexagonal net but not for a
p
(
2
×
2
)
overlayer, we look at the elementary excitation near a defect,
either an extra adatom or a missing one (see Fig.
2.2
). For just a right-triangle trio,
there are no such trios (no 2
E
RT
) in the excited state when there is a vacancy; when
there is an extra adatom, there are two RT trios in the ordered state which are lost
in hopping to the nearest neighbor (where another two RT trios occur). So in both
cases, there is no change in the number of RT trios, i.e., no change proportional
to
E
RT
is involved. A similar effect occurs with a linear trio, but with a different
(
2
×
2
)
