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**The Dynamics of Fullerene Structure Formation : Order
out of Chaos Phenomenon**

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**A. Mary Selvam and Suvarna Fadnavis**

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**Indian Institute of Tropical Meteorology,
Pune 411 008, India**

The *buckminsterfullerene* **C**_{60}
molecular structure is basically a three dimensional quasiperiodic *Penrose*
tiling pattern which incorporate the golden mean **t
=(1+Ö 5)2 @
1.618** in its geometry. More than a decade
after its discovery, we still lack a completely satisfying understanding
of the fundamental chemistry, that takes place during *fullerene*
formation. It is now felt that any reasonable reaction mechanism must include
a kinetics driven process that goes directly to *fullerene*, rather
than other carbon allotropes (J.W. Mintmire 1996: *Science* **272**,
45). **C**_{60} molecules form spontaneously during vaporization
of carbon associated with intense heating and turbulence such as in electrical
arcs or flames. Self-organization of fluctuations in the highly turbulent
(chaotic) atomized carbon vapor appears to result in the formation of the
stable structure of **C**_{60} and therefore may be visualized
as *order out of chaos* phenomenon. The geometry of **C**_{60},
namely, the selfsimilar quasiperiodic *Penrose* tiling pattern implies
long-range spatiotemporal correlations. Such nonlocal connections in space
and time are ubiquitous to dynamical systems in nature and is recently
identified as signatures of *self-organized criticality* (P.C. Bak
et. al. 1988: *Phys. Rev. A* **38**, 364). For example, atmospheric
flows exhibit long-range spatiotemporal correlations on all spatial and
temporal scales manifested as the *fractal* geometry to the global
cloud cover pattern concomitant with inverse power law form for power spectra
of temporal fluctuations documented and discussed in detail by Lovejoy
and his group (Y. Tessier *et al*. 1993: *J.Appl.Meteor*.**32(2)**
, 223). Mary Selvam (1990: *Can.J.Phys*. **68**, 831) and Selvam
and Fadnavis (1999: *Chaos, Solitons and Fractals* **10(8)** 1321):
http://xxx.lanl.gov/abs/chao-dyn/9806002
) have proposed a cell dynamical system
model for turbulent fluid flows where it is shown that the observed self-organized
criticality is a signature of quantum-like mechanics governing flow dynamics.
The model is based on the concept that large scale fluctuations can be
visualized as the integrated mean of enclosed small scale fluctuations.
Therefore the energy spectrum of fluctuations (eddies) follows the statistical
normal distribution. The square of the amplitude of eddy fluctuations represents
the probability density. Such a result that the additive amplitude of eddies,
when squared represent probability densities is observed in the subatomic
dynamics of quantum systems. Therefore macroscale turbulent fluid flows
follow quantumlike mechanical laws. The model also predicts an overall
logarithmic spiral trajectory for the flow pattern with the quasiperiodic
*Penrose* tiling pattern for the internal structure. It is proposed
that the intense turbulent fluctuations of vaporized carbon atoms may be
visualized as turbulent fluid flows with spontaneous formation of quasicrystalline
structure of the *Penrose* tiling pattern. The theoretical concept
enables to derive fundamental constants of atomic physics such as the *fine
structure constant* and the *ratio of electron to proton mass*
in terms of the universal *Feigenbaum*’s constants **a** and **d**.
The **C**_{60} structure formation is therefore a dynamical
process of self-organization of sub-quantum level turbulent (chaotic) fluctuations
leading to ordered structure formation and is directly related to the newly
emerging field of *nonlinear dynamics and chaos*. El Naschie (1997:
*Chaos, Solitons and Fractals* **8(11)** , 1873-1886) has shown
mathematically the *fractal* nature of space time fluctuations in
quantum systems.

Acknowledgements

The authors are grateful to Dr. A. S. R. Murty for
his keen interest and encouragement during the course of the study.