The structure of the damage region

The radius of gyration *R*_{n}, calculated as a function of the number
*N* of atoms knocked out (with initial kinetic energy of 416 eV) of
their lattice site into the same volume, as described above, is shown
in figure (9.7). It can be seen that with increasing number of
bombarding atoms the affected volume increases, however *R*_{n}approaches a saturation at about *R*_{max} =6.85 Å for twelve
bombarding atoms. This saturation was expected since the bombarding
atoms are directed toward a similar region in space. This prevent the
extension of the damage area by successive bombardments.

We are interested in analyzing separately the *core* of the
damage, defined to be the volume inside a sphere of radius *R*_{max}centered on
,
and its *periphery*, defined as the
volume outside this sphere, up to a radius of about 2*R*_{max}, since
these may contain different amounts and kinds of defects, and may thus
respond differently to annealing (as will be discussed in the next
section).

The percentage of atoms inside the core that are threefold and
fourfold coordinated has been calculated as a function of *N*, and the
results are shown in figure (9.8). The diamond lattice is, as
expected, destroyed by the bombarding atoms and as the number of
bombarding atoms increases the number of threefold coordinated atoms
increases at the expense of fourfold coordinated atoms. The structural
change to the diamond lattice can be further analyzed by studying of
the bond length distribution.

The radial distribution function *g*(*r*) is calculated as a function of
*N* for the atoms within the core and is shown in figure (9.9)
for selected values of *N*. It can be seen that the long range order
of the diamond lattice completely disappears for *N*=12, and only
short range order remains. The coordination number obtained by
calculating the area inside the first peak of *g*(*r*), (figure
(9.10)) decreases as *N* increases, consistently with the
amorphization observed. The first peak, that was centered at 1.545 Å
(which corresponds to the bond length in diamond) decreases and
broadens with increasing number of bombarding atoms, and its center
shifts toward the graphite bondlength (1.46 Å as calculated with
the Tersoff potential). In other words some bonds are shortened and
others are extended (see the insert in figure (9.9), that shows
the first peak of the radial distributions), spanning all lengths from
below the bond length in graphite, to above that of diamond. It should
also be noted that the unimodal distribution of the first peak of
*g*(*r*) for *N*=12 indicates that no distinct configuration
(graphite-like or diamond-like) dominates in the heavily damaged
region.

For *N*=12, the partial radial distribution function between threefold
coordinated atoms in the core, *g*_{33}, between threefold and
fourfold coordinated atoms, *g*_{34}, and for fourfold coordinated
atoms only, *g*_{44}, are calculated and are shown in figure
(9.11). One can see from the insert in figure
(9.11), in which the first peak of the partial distributions
is enlarged, that the bond length increases with the coordination
number. That is, the shortest bond lengths are those between threefold
coordinated atoms, and the longest bond lengths can be attributed to
bonds between fourfold coordinated atoms.

An important feature of our damage sample is the presence of a second
very sharp peak observed at about 2.1 Å in *g*_{33}. Since this
appears exclusively in *g*_{33}, one must attribute this peak to the
second nearest neighbors of threefold coordinated atoms. Such a short
second neighbor distance can exist only as the diagonal distance in
quadrilaterals. It will be shown that these quadrilateral structures
are highly unstable and almost entirely disappear upon annealing. It
has to be mentioned that this relatively short distance second peak
has been experimentally found in tetrahedral amorphous carbon by
Gilkes
[116], using high-resolution neutron scattering. It has also
been obtained by Marks *et. al.* [44] by *ab initio*
MD calculations.

In summary, the structure obtained after repeated bombardments of the
same region in the diamond lattice is heavily damaged, containing more
than 60 % of threefold coordinated atoms, yet with no distinct
diamond or graphite like character. Since the simulation is performed
at 0K, interstitial-vacancy recombinations due to some diffusional
process are inhibited after the cooling down of the thermal spike. The
rapid increase in *R*_{n} and the decrease in fourfold coordinated atoms
found here, as a function of the number of bombarding atoms, is thus
due to bond breakage induced by both the initial energetically
displaced atoms and by the recoiling "higher generation" atoms
involved in the damage cascade. As expected, the number of fourfold
coordinated atoms decreases by the bombardment process, while the
number of threefold coordinated atoms increases. The bombardment does
not only induce transformation of fourfold to threefold coordinated
atoms by bond breakage, but it also gives rise to the formation of
clusters of threefold coordinated atoms, as reflected by the increase
in *g*_{33}(*r*) as *N* increases. Indeed, in the core, each threefold
coordinated atom is found to be connected on the average to 1.74 other
threefold coordinated atoms, for *N*=12, while in the periphery, this
average is only 0.7. That is, the core of the damage contains a cluster
of connected threefold coordinated atoms, while in the periphery the
threefold coordinated atoms are isolated point defects.