Chemistry Reference
In-Depth Information
Table 3 Sphericities of common polyhedral
Volume
Surface area
Sphericity
2/12
s
3
3
s
2
Tetrahedron
0.671
√
√
s
3
6
s
2
Cube
0.806
2/3
s
3
3
s
2
Octahedron
2
0.846
√
√
3/4
s
3
3/2
s
2
+3
s
2
Trigonal prism
0.716
√
√
5)
s
3
/4
5)
s
2
Dodecahedron
(15 + 7
3
(25 + 10
0.910
√
√
√
2)
s
3
/3
3)
s
2
/12
Cuboctahedron
(5
(6 +
0.877
√
√
5)
s
3
/12
3
s
2
Icosahedron
5(3 +
5
0.939
√
√
5)
s
3
/12
3
s
2
Icosahedra sharing vertex
10(3 +
10
0.745
√
√
10(3 +
√
5)
s
3
/12
380/40
√
3
s
2
Icosahedra sharing face
0.890
3.760
s
3
12.99
s
2
Fused icosahedra
0.900
2.324
s
3
9.330
s
2
Bicapped pentagonal prism
0.909
4.044
s
3
14.330
s
2
Fused bicapped pentagonal prisms
0.857
Sphere 1.00
The calculations are based on polyhedra constructed from regular polygons with sides
¼
s
form the basis of infinite bulk structures, which are characterised by fcc, hexagonal
or bcc packing arrangements. Therefore, larger clusters with fivefold symmetry
must at some stage undergo a phase transition when they reach a certain size in
order to achieve infinite close packing. The precise size at which this transformation
occurs remains the subject of intense study and debate [
98
]. The sphericity index
may also be used to quantify the distortions in oblate, prolate and toroidal gold
clusters, which have been discussed above.
Close-packed structures with high symmetries, i.e. T
d
,O
h
, and I
h
[
84
], may be
generated, but once again they do not necessarily have high sphericities. For
example, tetrahedral close-packed arrangements of metals with 1/6{
k
(
k
+1)
(
k
+ 2)} (
k
¼ 2, 3, 4, etc. ¼ number of atoms on equivalent edges) atoms may be
constructed, but their sphericities (0.671) deviate greatly from the spherical ideal.
The icosahedron and cuboctahedron which have higher sphericity indices provide a
better basis for constructing clusters which are simultaneously close packed and
approximately spherical. Related formulae for the cube and octahedron are
summarised below. Close-packed structures based on fcc packing with high sym-
metry polyhedra depends on the number of atoms on equivalent edges (
k
)
k
2
3
4
5
N
Tetrahedron
4
10
20
35
1/6{
k
(
k
+ 1)(
k
+ 2)}
1/3{
k
(2
k
2
+ 1)}
N
Octahedron
6
19
44
85
k
{4
k
2
N
Cube
14
63
172
465
6
k
+3}
For 12 vertex polyhedra, e.g. icosahedra, decahedra and cuboctahedra, with
K
concentric shells, the total number of atoms
N
is given by
:
310
K
3
þ
15
K
2
NðÞ
¼1
=
þ
11
K þ
3
This corresponds to
N
¼ 13, 55, 147, 309, 561, 923, 1,415, 2,057, etc.
