Page 180 - Carbon Nanotubes

et
172 U. ZIMMERMAN al.
later, does not apply in our case. In the following one-tenth of a turn (36") around the 5-fold axis
paragraphs we will often specify the positions of the through its center. The coordination to the atoms of
metal atoms relative to the central CW molecule. This the first layer will then be only two-fold, but the cov-
is done for clarity and is not meant to imply any di- erage will be quite even, making the latter of these two
rect interaction between the c60 and the atoms of the structures the more probable one.
second layer. The latter structure could be described as an 'edge-
In constructing the second layer, it seems reason- truncated icosahedron' with 20 triangular faces, each
able to expect this layer to preserve some of the char- face consisting of the three atoms at the icosahedral
acteristic symmetry elements of the first layer (Le., the vertices with a smaller, almost densely packed trian-
fivefold axes). The second layer on c60 contains 72 gle of three atoms set in between (exemplarily, one of
atoms, a number being indivisible by 5. This requires these triangles has been shaded). Note that this layer,
that each of the five-fold symmetry axes passes having no atoms right on the edges, is not identical to
through two metal atoms. Consequently, in the sec- a Mackay icosahedron[l6] which is formed by pure al-
ond layer there must be one metal atom situated above kaline earth metal clusters[lO,l l]. However, in this
each of the 12 pentagonal faces of c60. Let us first structure the two rows of atoms forming the truncated
assume that the second layer has the full icosahedral edges are not close-packed within the layer. This might
symmetry I,, of the first layer. The remaining 60 at- be a hint that with the structure depicted on the up-
oms may then be arranged basically in two different per right in Fig. 4 we have not yet found the most sta-
ways. The first would be to place the atoms such that ble configuration of the second layer.
they are triply coordinated to the atoms of the first Up to this point, we have assumed that the second
layer (i.e., placing them above the carbon atoms of the layer of atoms preserves the full symmetry (Ih) of
C6, molecule as shown in Fig. 4 on the upper left). the fullerene inside. Let us now allow the second layer
The atoms above the pentagons of c60 (black) consti- to lower its symmetry. This can be done in a simple
tute the vertices of an icosahedron, the other atoms way: model the interaction between metal atoms by a
(white) resemble the C,,-cage. This structure can also short-range pair potential with an appropriate equi-
be visualized as twelve caps, each consisting of a librium distance and let the atoms of the second layer
5-atom ring around an elevated central atom, placed move freely within this potential on top of the first
at the vertices of an icosahedron. This structure, how- layer. This allows the atoms to move to more highly
ever, does not result in an even coverage: there are 20 coordinated positions. Starting with atoms in the ar-
large openings above the hexagonal faces of Cm while rangement with Ih-symmetry, the layer will relax
neighboring caps overlap above the double bonds of spontaneously by rotating all 20 triangular faces of at-
C,,. Pictured on the upper right in Fig. 4 is a second oms around their three-fold axes by approximately
way to arrange the 60 atoms with Ih symmetry, ob- 19". The resulting structure is shown at the bottom of
tained by rotating each of the caps described above by Fig. 4. One of the rotated triangles has been shaded
and the angle of rotation marked. In a projection on
a plane perpendicular to the threefold axis, each pair
of atoms at the edges of the triangle lie on a straight
line with one of the three atoms on the surrounding
icosahedral vertices. The two rows of atoms along the
former truncated edges have now shifted by the radius
of one atom relative to each other in direction of the
edge, leading to close packing at the edges. Of course,
the triangles could have been rotated counterclockwise
by the same angle, resulting in the stereoisomer of the
structure described above. This structure no longer has
Ih-symmetry. There are no reflection planes and no
inversion symmetry. Only the two-, three-, and five-
fold axes remain. The structure belongs to the point
group I (order 60). I is the largest subgroup of I,,.
The layer has, thus, undergone the minimum reduc-
tion in symmetry.
Of the three arrangements of atoms in the second
layer shown in Fig. 4, we find the one on the bottom
(symmetry I) the most probable. It optimizes the co-
ordination of neighboring atoms within the layer and,
Fig. 4. Three possible geometries for arranging the 72 atoms as we will see further down, this arrangement can also
of the second layer: the atoms above the pentagons of Cs0 be well extended to C,, coated with metal.
are shaded. The structure on the upper left can be trans- Of course, after having observed two complete layers
formed into the more evenly distributed arrangement of of metal around a fullerene, we searched for evidence
atoms on the upper right by 36" turns of the caps around the
five-fold axes. From this, the structure on the bottom can be for the formation of additional layers. However, be-
obtained by rotating each triangular face of atoms by 19". fore looking at experimental data, let us try to con-