Chemistry Reference
In-Depth Information
An analogous formula exists for bbc close-packed structures based on the
14 vertex rhombic dodecahedron:
NðÞ
¼4
K
3
þ
6
K
2
þ
4
K þ
1
:
This corresponds to
N
¼ 15, 65, 175, 369, 671, 1,105, 1,695, 2,465, etc.
The number of atoms in the resultant close-packed arrangement may be
increased by adding capping atoms or decreased by truncation. However, if the
symmetry of the central polyhedron is to be maintained, the resulting number of
capping or truncated atoms has to reflect their positions relative to the symmetry
elements.
If attention is focussed on T
d
,O
h
and I
h
structures, then the point group
symmetries may be used to establish the number of atoms either in general or
special positions lying on symmetry elements. For example, for M
20
the following
permutations are the only ones allowed. The local symmetries of the atoms are
indicated in brackets:
T
d
4C
3
ðÞþ
4
C
3v
þ
12
Cs
O
h
6C
4
ðÞþ
6
C
4v
þ
8
C
3v
12 C
2
ðÞþ
8
C
3v
I
h
20 C
3
ðÞ
The resultant arrangements are composite structures based on tetrahedra (C
3v
),
octahedra (C
4v
), cubes 8(C
3v
) and cuboctahedra 12(C
2v
). The most stable close-
packed structure is one with the maximum number of nearest neighbours and has
successive layers of atoms in complementary positions. On the other hand the most
spherical structure is that which has the maximum number of atoms on the surface
layer, and this is achieved for evenly distributed arrangements. Of the M
20
struc-
tures, the first choice is preferred since it satisfies the requirements of complemen-
tary angular coordinates on successive layers. In the second possibility, the location
of atoms on two sets of C
3v
special positions leads to a structure which has atoms
placed above each other. The third and fourth structures cannot lead to close-packed
structures because there are too many atoms on one layer (12 or 20) or a cube is
generated as the central moiety. These geometric constraints have been discussed in
more detail by Lin and Mingos, who also noted coincidences between the geometric
and jellium electronic shell structures [
84
].
In recent years the determination of an increasing number of clusters by crys-
tallographic techniques has provided additional experimental evidence regarding
the preferred close-packed and spherical arrangements, and Dahl and his
co-workers and Longoni and Iapalucci have provided detailed reviews and discus-
sions of cluster structures based on icosahedral packing modes [
119
-
124
].
