8.3 The Lesson of Sodium Chloride 189
(a) (b) (c) (d)
Figure 8.6 The B
¨
urger mechanism in terms of periodic sur-
faces. (a) P* and (d) F* surface representing the B2 and B1
structures, respectively, (b) and (c) are intermediate configu-
rations.
As the PNS separates positive and negative ‘‘charges’’ from each other, for
particular isovalues of the starting function, f
A
(x, y, z), the surface is a collection
of bubbles enclosing positive and negative charges, respectively. The weighting
factors w
A
and w
B
are chosen such that the bubbles remain disconnected for each
value of s. Variation of the mixing factor, s, results in a concerted atomic movement,
periodic at each stage, and continuous in the atomic displacements. The degree
of mixing of the limiting functions may be interpreted in terms of a reaction
coordinate. For each value of s ∈ [0, 1], a different configuration of the atoms
is connected. The cell setting can be chosen rhombohedral (hexagonal setting),
applying the transformation matrix (101,
111, 011) to the cubic NaCl cell. The set
(111), eight vectors, splits into (101), six vectors, and (003), two vectors. The (101)
set alone generates the topology of the P* surface. Both surfaces collapse at sites
(0, 0, 0) and (1/2, 0, 0), and the only free parameter is the ratio c/a, which can be
used as reaction coordinate. This reproduces the B
¨
urger model (Figure 8.6).
Transformation of the cubic reflex sets into a common orthorhombic cell is the
starting point for the description of the Hyde and O’Keeffe model in terms of
periodic surfaces. The origin of the F cell has to be shifted by (
1
4
,
1
4
, 0), whereby the
phases of the vectors (111) and (11
1) change from 0 to π. The origin of the P cell
is shifted by (0,
1
2
, 0), which only affects the phase of reflex (010, α
010
= π). The
surface describing the NaCl phase collapses around sites (3/4, 1/4, 1/4) and (1/4,
1/4, 1/4) for positive or negative choices of the isovalue, respectively, while the sites
are (1/4, 1/4, 0) and (3/4, 1/4, 1/2) for the CsCl-type structure. The linear mixing
results in a continuous and synchronous movement of the Na
+
and Cl
−
sites as
a function of the degree of mixing, s. Atoms in adjacent (100)
NaCl
layers displace
along [110] in an antiparallel fashion. Hereby each atom undergoes a displacement
of 1/8 of the face diagonal of the NaCl unit cell (Figure 8.7).
The modeling approach defines a one-dimensional collective coordinate, s.For
each path the transition states correspond to a single value s
TS
. We refer to
the transition state as the configuration along the collective coordinate s,which
equals probability of relaxing toward either NaCl or CsCl. If p(NaCl) represents
the probability of forming NaCl, the expression p
NaCl
= 0.5 = p
CsCl
defines the
transition state configuration. To derive the transition state, configurations obtained