Fractal dimension of collision cascades
F. Kun
*
and G. Bardos
Department of Theoretical Physics, Kossuth Lajos University, P.O. Box 5, H-4010 Debrecen, Hungary
Received 6 March 1996; revised manuscript received 11 November 1996
The geometrical structure of the vacancy distribution in collision cascades is studied using Monte Carlo
simulations. Based on a Flory–de Gennes-type approach, a relation of the fractal dimension, the self-similarity
dimension, and the dimension of the embedding space is established. It is shown that, varying the parameter of
the interaction potential, a structural transition takes place in the cascade from an open branching structure to
a space-filling one. Based on the results the spike condition of Cheng, Nicolet, and Johnson Phys. Rev. Lett.
58, 2083 1987 is revisited. S1063-651X9712602-2
PACS numbers: 05.90.+m
I. INTRODUCTION
Collision cascades develop in condensed matter as a con-
sequence of irradiation with energetic beams of particles.
The bombarding particles transfer their kinetic energy in se-
ries of collisions with the target atoms and the energized,
recoiling atoms generate further recoils in their own slowing-
down process. The result of this energy sharing process is a
collision cascade.
To study the geometrical structure of collision cascades
there are two possible viewpoints: On the one hand, the cas-
cade can be considered as a treelike geometrical object,
which is composed of the trajectories of the moving particles
and points where the collisions occurred 1,2. Recently we
showed that in this consideration the cascade-tree exhibits
multiscaling and multifractality, which is a direct conse-
quence of the underlying multiplicative process of the cas-
cade mechanism 3.
On the other hand, the cascade can be treated as a
branched aggregate of the vacancies created in sequential
collisions during the cascade evolution. This vacancy distri-
bution in the target is bounded by the interstitials, making
this damaged region in the solid well defined. Randomly
branched aggregates occur in many physical systems such as
branched polymers, the sol-gel transition, percolation, turbu-
lence, nucleation, the formation of smoke particles, and elec-
tric breakdown. The common feature of these objects is that
they all show a strong degree of self-similarity 4. From the
geometrical point of view the structure of the vacancy distri-
bution in the collision cascades is analogous to the structure
of randomly branched aggregates.
Recently vacancy distribution in collision cascades has
been investigated from the viewpoint of fractal geometry by
means of analytical calculations and of Monte Carlo MC
simulations in the framework of the binary collision approxi-
mation BCA. These investigations have been extended to
the study of the self-similarity properties of the cascade
5,6, to the determination of its fractal dimension for differ-
ent interaction potentials 1–3,7,8, and to the study of
cascade-subcascade transition and spikes 5,8. A simple de-
terministic fractal-tree model see Fig. 1 was proposed 5
as an average cascade for the case of an inverse-power po-
tential of the type
V r =G m r
-1/m
, 0 m 1. 1
It was shown that the self-similarity dimension of the deter-
ministic tree D
0
( m ) =1/2m in d =3 embedding Euclidean
space. It depends solely on the parameter m of the interaction
potential. To test the predictions of this model, MC simula-
tions were performed. It was found that the measured MC
*Electronic address: feri@dtp.atomki.hu
FIG. 1. Deterministic cascade trees up to ten generation steps
with different similarity ratios .( is the ratio of two successive
branches in the tree.a =0.6. b =0.7. One can observe that
for increasing decreasing m) the overlap of different branches is
increasing.
PHYSICAL REVIEW E FEBRUARY 1997 VOLUME 55, NUMBER 2
55 1063-651X/97/552/15086/$10.00 1508 © 1997 The American Physical Society