- Email: [email protected]
Quantum aspects of fractal structures and monomers
Chaos, Solitons and Fractals 14 (2002) 817–822 www.elsevier.com/locate/chaos
Quantum aspects of fractal structures and monomers R.J. Slobodrian D epartement de Physique, Universit e Laval, Ste. Foy, Quebec, QC, Canada G1K 7P4 Accepted 15 January 2002
Abstract Structures with fractal geometry and discrete units, albeit complex, pervade our universe, from subatomic systems to galaxies. Randomness and discreteness are dominant characteristics. It is possible to surmise a common ground between quantum and fractal physics. Some examples will be discussed in detail. Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction The definition of fractals based upon the concept of self-similarity is currently accepted [1,2]. This definition encompasses mathematical (deterministic) as well as nature’s fractals. They constitute topological spaces in the general sense, where a metric can be introduced. Fractals are characterized by generalized dimensions [3] which are usually noninteger and always different from the topological dimension of the embedding space (which are integer: 1 for a line, 2 for a plane, etc.) [4,5]. Recently fractality concepts have been applied to non-static systems [6] implying new perspectives for dynamical applications to subatomic particle systems in swift motion, belonging to the domain of quantum mechanics and beyond.
2. Structures of aggregates and monomers A special effort has been devoted in recent years to fractals embedded in three-dimensional space [7], produced by evaporation–condensation of metals within an inert gas atmosphere [8]. Fig. 1 shows images of aggregates of Bi at different levels of statistics observed with a scanning electron microscope (SEM). Spheroidal units known as monomers with varying radii are clearly seen. These aggregates are particularly solid because the monomers are not in simple contact but are connected by fusion along the tangential planes (see Fig. 2 of [9]). Laser beam scattering techniques have allowed non-destructive proof of fractality and the determination of fractal dimensions [10] of such aggregates. Computer simulations based on the diffusion limited aggregation model (DLA) agree well qualitatively with the observed aggregates [5], see Fig. 2. Randomness is a salient characteristic apparently well reproduced by Brownian motion patterns in the displacement and attachment of monomers. This motion is reminiscent of quantum random paths like the Schr€ odinger ‘‘Zitterbewegung’’ (zbg) of electron motion. There is a close relationship between quantum mechanics and Brownian motions, noted by several authors [4], due to the similarity of the diffusion and the Schr€ odinger equation as well as in the actual paths of ‘‘quantum’’ particles and Brownian paths. Relativistic quantum mechanics in the context of a finite zbg model has been explored in the recent past by Pierre Noyes [11], with surprising agreement with experiments, demonstrating again the close relationship of quantum and other random motions. Monomers are not always spherical but show also crystalline structures at micrometer to nanometer scales, an example is shown in Fig. 3. Aggregates in general are not completely characterized by the fractal dimension. The
E-mail address: [email protected] (R.J. Slobodrian). 0960-0779/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 0 2 ) 0 0 0 2 6 - 7
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R.J. Slobodrian / Chaos, Solitons and Fractals 14 (2002) 817–822
Fig. 1. (a) General view of a Bi aggregate. Notice the variable diameters of the monomers. (b) Magnified view of part of (a). Notice the self-similarity property evident in the sets of much smaller monomers, with respect to the aggregates of bigger ones.
criterion of similarity between aggregates has been discussed before [7]. A minimal additional requirement for characterization is the specific metric of the aggregate, i.e., distance within its space from point A to point B, which is clearly non-Euclidean and may vary from aggregate to aggregate. Another type of metallic aggregate is drastically different from the ones discussed above, it is fern-like and shows a higher degree of symmetry. It will be discussed elsewhere [12] in detail, classical thermodynamics considerations seem to explain their behaviour.
3. Quantum physics of metallic monomers The monomers of metallic aggregates are also known as metal clusters. Recent progress in their understanding has been made since the discovery of maxima in the abundances of metal clusters by Knight et al. [13] in a beam of alkali atoms, with ‘‘magic numbers’’ of valence electrons following the sequence 2, 8, 40, 58, . . . which could be associated with filled spherical shells in an average potential, like that of the nuclear optical model. Clemenger [14] has used the Nilson nuclear model without spin–orbit interaction, as no evidence was found for the latter.
R.J. Slobodrian / Chaos, Solitons and Fractals 14 (2002) 817–822
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Fig. 2. Computer simulated aggregate.
Fig. 3. Cr aggregate with crystalline monomers (mostly cubic).
This subject has grown in complexity and is covered extensively in review articles by Brack [15] and de Heer [16]. The initial observation took place on so-called one-electron atoms (alkaline), but the idea can be extended in general to metallic atoms, which have rather loose electrons. Periodicities of abundances were observed for clusters consisting of 1000 atoms and beyond. This means sizes in excess of 10 nm for medium weight metallic atoms. Fig. 4 shows an image of a Zn monomer with an overall size of 500 nm and in the enlargement it is possible to see substructures down to about 10 nm. These images were obtained with a Hitachi STM 1329 6 electron microscope allowing to observe samples without the need to evaporate a conducting coating on them, thus they are totally undistorted. One can surmise that the monomers are an extension of the clusters beyond the ‘‘magic numbers’’ cited above. It is clear that the monomers constitute a bridge spanning the typical quantum mechanical scales and the semiclassical to classical scales. In the case of aggregates and monomers of our experiments X-ray analyses have shown that there is always an underlying nano-crystalline substructure. This fact allows to formulate an alternative model for metallic clustering based on the nuclear alpha particle model [17], which competed quite successfully with the single particle shell model and its variations. The alpha particle is very tightly bound and pictured as a tetrahedron, in fact, crystal-like. Quantum mechanically, there is a sequence of alpha particle nuclei and additional nucleons are coupled to such ‘‘cores’’. The basic idea for monomers is that metallic atoms form units determined by minimum energy configurations, for example, cubic systems at subnanometer scales, and these attach themselves into bigger units yielding finally either spherical or crystalline monomers, which then
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R.J. Slobodrian / Chaos, Solitons and Fractals 14 (2002) 817–822
Fig. 4. (a) Complex monomer of Zn with apparent substructure. (b) Magnified view of a section of (a) showing structures down to the 10 nm scale, already realm of quantum phenomena.
aggregate into fractals. This would be consistent with the X-ray observations cited earlier. Complex associations of atoms occur at low levels, as has been demonstrated by the carbon fullerenes [18], hence ‘‘a fortiore’’, simpler crystals, energy favored, should form preferentially upon condensation of vapors over the accumulation of atoms onto a spherical surface. In large numbers the appearance – at micrometer and larger sizes – may be spherical or crystalline. Summarizing this section, the monomeres have their origin in small associations of atoms governed by quantum Hamiltonians and wave equations. Fig. 4 shows indeed surface undulations related to such small associations of atoms. It is suggested here that the units of complex associations may be actually crystals and not the atoms themselves. A transition does occur from such dimensions to micrometer and above scales, towards semiclassical and classical properties. This is far from trivial because in that passage there is a fundamental change of the quantum descriptions in coordinate space or, equivalently in momentum space, towards a full Hamiltonian ðpi ; qi Þ phase space description or a Lagrangian ðqi ; qi Þ mapping.
4. Quantized subatomic systems: fractal characteristics The units of these systems are in permanent motion but it is possible to formulate a model with an image in configuration space [19] having heuristic content. In particular, a two-dimensional image of the uranium nucleus was
R.J. Slobodrian / Chaos, Solitons and Fractals 14 (2002) 817–822
821
Fig. 5. A fractal configuration of U electrons.
obtained. It has been shown recently that such images are fractal [6] and that remarkably similar images can be generated with stochastic simulation techniques also used for aggregates. This is not surprising because quantum states and their vectors have also a random behaviour. The first subatomic components are electrons and, for example, the swarm of electrons of the uranium atom has also been obtained with the simulation techniques cited and its two-dimensional image is shown in Fig. 5. It is clear that force fields and exclusion criteria should still be included in this alternative approach, but the image shows that simple stochastic principles yield a plausible first approximation to a configuration of the ‘‘instantaneous’’ positions. In the same order of ideas one may extend fractality to subnucleon systems. Cohen [19] has produced images of such systems based on elementary particle physics principles. There is a remarkable self-similarity in some of the subnucleon systems, for example, the hadronic and leptonic multiplets. Heisenberg predicted that as more energy is pumped into the center of mass of a colliding system an increasing number of particles will emerge from the collision, including ‘‘new’’ ones. Of course, such colliding systems at energies considerably higher than the rest masses of nucleons does not provide much information on what one may call the ‘‘static proton’’, yet the physics is interesting and the successive ‘‘new particles’’, very ephemeral, seem to indicate a natural tendency to self-replication, with self-similar systems but with ever shorter lifetimes.
5. Discussion The mathematics of physics is complicated. The ideal point particles do not exist except perhaps in the case of leptons. The apparent regularity and continuity of the macroscopic world is rapidly destroyed going to the microscopic particles and their constituents. In addition the systems of concern are examples of N-body systems [20], where in fact N tends to infinity, and simplifying assumptions are inevitable. As mentioned above there is a drastic passage of the mathematics and spaces required in the transition from the classical to the quantum description. Borderline systems are amenable to so-called ‘‘semiclassical physics’’ treatment [21] using quantum thumb rules or Sommerfeld path integrals [22]. The fractals shown in Figs. 1–4 cover the scales of the transition and exhibit persistent randomness which is rooted in fluctuations of quantum nature, at the level of the monomers’ components and the attachment processes. At the other end of the universal scale galaxies (and their stars) exhibit similar fractal structures dominated by randomness [23], once again evidence of the existence of fundamental natural mechanisms at all scales. For example, reducing the spheres (representing electrons) to points in Fig. 5 one can obtain qualitatively the image of a galaxy of stars, or a section of our own galaxy.
6. Concluding remarks There is an undeniable relation between fractal and quantum physics like the aggregation of atoms and quantum paths and interactions. The isomorphism of the mathematics describing physical phenomena from widely different fields, like for example, the application of BCS techniques of superconductivity to the realm of nuclear physics or the Brownian motion–Schr€ odinger equations similarity, may indicate the existence of a possible unifying basis for fractality
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and quantization, to be much more fundamental and far reaching. There is a vast difference in scale between subatomic systems and the typical fractal aggregates, but fractal properties have been shown to be present even in quantum systems having permanent internal motion for their ‘‘instantaneous’’ configurations. This indicates that time dependent quantized systems persist in having fractal properties down to very small space-time scales. Continuity and smoothness at macroscopic scales of media and force fields is a misleading illusion. It comes about due to the statistics of large number of events and may be a reasonable approximation, but nature is basically discrete and stochastic. The starting point of quantum mechanics with differential equations, continuity conditions and Hilbert spaces may have to be replaced by an approach based on discretness and randomness, for space-time, particles, charges and force fields alike.
Acknowledgements The points discussed in this paper correspond to work and experience gathered during a period of some 12 years on fractals, mostly based on experimental evidence. Many scientists and technicians have contributed to the success of this research and it would be too long to cite them all here. However it is my pleasure to thank warmly Dr. Claude Rioux who has carried out much of the work in low gravity for his unfailing dedication and competence throughout all these years. The support of the Canadian Space Agency was essential to the experimental programme. Universite Laval has also contributed significantly with infrastructure and technical support.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
Mandelbrot B. Les objets fractals, forme, hasard et dimension. 3rd ed. Paris: Flammarion; 1989. Feder J. Fractals. New York: Plenum Press; 1988. Edgar GA, editor. Classics on fractals. Reading, MA: Addison Wesley; 1993. Kr€ oger H. Phys Rep 2000;323:81, and Refs. therein. This work reviews in depth fractal geometry relations with quantum physics. Slobodrian RJ. Phys Can 1995;51:245, and Refs. therein. Leclerc J-C, Rioux C, Slobodrian RJ. Chaos, Solitons & Fractals 2001;12:97. Gauthier M, Cossette M, Rioux C, Slobodrian RJ. Chaos, Solitons & Fractals 1993;3:687. Slobodrian RJ, Rioux C, Leclerc J-C. In: Sch€ umann B, editor. First International Symposium in Mycrogravity Research and Applications in Physical Sciences and Biotechnology, Vol. II. Noordwijk: ESTEC, ESA Publ. 2001; p. 779. Slobodrian RJ, Deladurantaye P, Gauthier M, Rioux C, Potvin L. Chaos, Solitons & Fractals 1996;7:25. Deladurantaye P, Rioux C, Slobodrian RJ. Chaos, Solitons & Fractals 1997;8:1693. Pierre Noyes H. SLAC-PUB-5189; 1990. Leclerc J-C. Proceedings of Spacebound 2000, May 2000, Available on CD-rom from Canadian Space Agency. Knight WD, Clemenger K, de Heer WA, Saunders WA, Chou MY, Cohen ML. Phys Rev Lett 1984;52:2141. Clemenger K. Phys Rev B 1985;32:1359. Brack M. Rev Mod Phys 1993;65:677. de Heer WA. Rev Mod Phys 1993;65:611. McCarthy IE. Introduction to nuclear theory. New York: Wiley; 1968. Baggot J. Perfect symmetry. UK: Oxford University Press; 1994. Cohen E. Comput Phys Commun 1992;70:441; Cohen E. Nucl Phys News 1992;5:20. Slobodrian RJ. Phys Rev C 1989;39:1052, and Refs. therein. Brack M, Bhaduri RK. Semiclassical physics. Reading, MA: Addison Wesley; 1997. Sommerfeld A. Phys Z 1911;12:1057, and Refs. therein. ec, Essiambre R.-J. Geometrie Fractale et Structure @ Grande echelle de l‘Univers, Memoire de ma^itrise. Universite Laval, Qub Canada; 1990.
Quantum aspects of fractal structures and monomers R.J. Slobodrian D epartement de Physique, Universit e Laval, Ste. Foy, Quebec, QC, Canada G1K 7P4 Accepted 15 January 2002
Abstract Structures with fractal geometry and discrete units, albeit complex, pervade our universe, from subatomic systems to galaxies. Randomness and discreteness are dominant characteristics. It is possible to surmise a common ground between quantum and fractal physics. Some examples will be discussed in detail. Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction The definition of fractals based upon the concept of self-similarity is currently accepted [1,2]. This definition encompasses mathematical (deterministic) as well as nature’s fractals. They constitute topological spaces in the general sense, where a metric can be introduced. Fractals are characterized by generalized dimensions [3] which are usually noninteger and always different from the topological dimension of the embedding space (which are integer: 1 for a line, 2 for a plane, etc.) [4,5]. Recently fractality concepts have been applied to non-static systems [6] implying new perspectives for dynamical applications to subatomic particle systems in swift motion, belonging to the domain of quantum mechanics and beyond.
2. Structures of aggregates and monomers A special effort has been devoted in recent years to fractals embedded in three-dimensional space [7], produced by evaporation–condensation of metals within an inert gas atmosphere [8]. Fig. 1 shows images of aggregates of Bi at different levels of statistics observed with a scanning electron microscope (SEM). Spheroidal units known as monomers with varying radii are clearly seen. These aggregates are particularly solid because the monomers are not in simple contact but are connected by fusion along the tangential planes (see Fig. 2 of [9]). Laser beam scattering techniques have allowed non-destructive proof of fractality and the determination of fractal dimensions [10] of such aggregates. Computer simulations based on the diffusion limited aggregation model (DLA) agree well qualitatively with the observed aggregates [5], see Fig. 2. Randomness is a salient characteristic apparently well reproduced by Brownian motion patterns in the displacement and attachment of monomers. This motion is reminiscent of quantum random paths like the Schr€ odinger ‘‘Zitterbewegung’’ (zbg) of electron motion. There is a close relationship between quantum mechanics and Brownian motions, noted by several authors [4], due to the similarity of the diffusion and the Schr€ odinger equation as well as in the actual paths of ‘‘quantum’’ particles and Brownian paths. Relativistic quantum mechanics in the context of a finite zbg model has been explored in the recent past by Pierre Noyes [11], with surprising agreement with experiments, demonstrating again the close relationship of quantum and other random motions. Monomers are not always spherical but show also crystalline structures at micrometer to nanometer scales, an example is shown in Fig. 3. Aggregates in general are not completely characterized by the fractal dimension. The
E-mail address: [email protected] (R.J. Slobodrian). 0960-0779/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 0 2 ) 0 0 0 2 6 - 7
818
R.J. Slobodrian / Chaos, Solitons and Fractals 14 (2002) 817–822
Fig. 1. (a) General view of a Bi aggregate. Notice the variable diameters of the monomers. (b) Magnified view of part of (a). Notice the self-similarity property evident in the sets of much smaller monomers, with respect to the aggregates of bigger ones.
criterion of similarity between aggregates has been discussed before [7]. A minimal additional requirement for characterization is the specific metric of the aggregate, i.e., distance within its space from point A to point B, which is clearly non-Euclidean and may vary from aggregate to aggregate. Another type of metallic aggregate is drastically different from the ones discussed above, it is fern-like and shows a higher degree of symmetry. It will be discussed elsewhere [12] in detail, classical thermodynamics considerations seem to explain their behaviour.
3. Quantum physics of metallic monomers The monomers of metallic aggregates are also known as metal clusters. Recent progress in their understanding has been made since the discovery of maxima in the abundances of metal clusters by Knight et al. [13] in a beam of alkali atoms, with ‘‘magic numbers’’ of valence electrons following the sequence 2, 8, 40, 58, . . . which could be associated with filled spherical shells in an average potential, like that of the nuclear optical model. Clemenger [14] has used the Nilson nuclear model without spin–orbit interaction, as no evidence was found for the latter.
R.J. Slobodrian / Chaos, Solitons and Fractals 14 (2002) 817–822
819
Fig. 2. Computer simulated aggregate.
Fig. 3. Cr aggregate with crystalline monomers (mostly cubic).
This subject has grown in complexity and is covered extensively in review articles by Brack [15] and de Heer [16]. The initial observation took place on so-called one-electron atoms (alkaline), but the idea can be extended in general to metallic atoms, which have rather loose electrons. Periodicities of abundances were observed for clusters consisting of 1000 atoms and beyond. This means sizes in excess of 10 nm for medium weight metallic atoms. Fig. 4 shows an image of a Zn monomer with an overall size of 500 nm and in the enlargement it is possible to see substructures down to about 10 nm. These images were obtained with a Hitachi STM 1329 6 electron microscope allowing to observe samples without the need to evaporate a conducting coating on them, thus they are totally undistorted. One can surmise that the monomers are an extension of the clusters beyond the ‘‘magic numbers’’ cited above. It is clear that the monomers constitute a bridge spanning the typical quantum mechanical scales and the semiclassical to classical scales. In the case of aggregates and monomers of our experiments X-ray analyses have shown that there is always an underlying nano-crystalline substructure. This fact allows to formulate an alternative model for metallic clustering based on the nuclear alpha particle model [17], which competed quite successfully with the single particle shell model and its variations. The alpha particle is very tightly bound and pictured as a tetrahedron, in fact, crystal-like. Quantum mechanically, there is a sequence of alpha particle nuclei and additional nucleons are coupled to such ‘‘cores’’. The basic idea for monomers is that metallic atoms form units determined by minimum energy configurations, for example, cubic systems at subnanometer scales, and these attach themselves into bigger units yielding finally either spherical or crystalline monomers, which then
820
R.J. Slobodrian / Chaos, Solitons and Fractals 14 (2002) 817–822
Fig. 4. (a) Complex monomer of Zn with apparent substructure. (b) Magnified view of a section of (a) showing structures down to the 10 nm scale, already realm of quantum phenomena.
aggregate into fractals. This would be consistent with the X-ray observations cited earlier. Complex associations of atoms occur at low levels, as has been demonstrated by the carbon fullerenes [18], hence ‘‘a fortiore’’, simpler crystals, energy favored, should form preferentially upon condensation of vapors over the accumulation of atoms onto a spherical surface. In large numbers the appearance – at micrometer and larger sizes – may be spherical or crystalline. Summarizing this section, the monomeres have their origin in small associations of atoms governed by quantum Hamiltonians and wave equations. Fig. 4 shows indeed surface undulations related to such small associations of atoms. It is suggested here that the units of complex associations may be actually crystals and not the atoms themselves. A transition does occur from such dimensions to micrometer and above scales, towards semiclassical and classical properties. This is far from trivial because in that passage there is a fundamental change of the quantum descriptions in coordinate space or, equivalently in momentum space, towards a full Hamiltonian ðpi ; qi Þ phase space description or a Lagrangian ðqi ; qi Þ mapping.
4. Quantized subatomic systems: fractal characteristics The units of these systems are in permanent motion but it is possible to formulate a model with an image in configuration space [19] having heuristic content. In particular, a two-dimensional image of the uranium nucleus was
R.J. Slobodrian / Chaos, Solitons and Fractals 14 (2002) 817–822
821
Fig. 5. A fractal configuration of U electrons.
obtained. It has been shown recently that such images are fractal [6] and that remarkably similar images can be generated with stochastic simulation techniques also used for aggregates. This is not surprising because quantum states and their vectors have also a random behaviour. The first subatomic components are electrons and, for example, the swarm of electrons of the uranium atom has also been obtained with the simulation techniques cited and its two-dimensional image is shown in Fig. 5. It is clear that force fields and exclusion criteria should still be included in this alternative approach, but the image shows that simple stochastic principles yield a plausible first approximation to a configuration of the ‘‘instantaneous’’ positions. In the same order of ideas one may extend fractality to subnucleon systems. Cohen [19] has produced images of such systems based on elementary particle physics principles. There is a remarkable self-similarity in some of the subnucleon systems, for example, the hadronic and leptonic multiplets. Heisenberg predicted that as more energy is pumped into the center of mass of a colliding system an increasing number of particles will emerge from the collision, including ‘‘new’’ ones. Of course, such colliding systems at energies considerably higher than the rest masses of nucleons does not provide much information on what one may call the ‘‘static proton’’, yet the physics is interesting and the successive ‘‘new particles’’, very ephemeral, seem to indicate a natural tendency to self-replication, with self-similar systems but with ever shorter lifetimes.
5. Discussion The mathematics of physics is complicated. The ideal point particles do not exist except perhaps in the case of leptons. The apparent regularity and continuity of the macroscopic world is rapidly destroyed going to the microscopic particles and their constituents. In addition the systems of concern are examples of N-body systems [20], where in fact N tends to infinity, and simplifying assumptions are inevitable. As mentioned above there is a drastic passage of the mathematics and spaces required in the transition from the classical to the quantum description. Borderline systems are amenable to so-called ‘‘semiclassical physics’’ treatment [21] using quantum thumb rules or Sommerfeld path integrals [22]. The fractals shown in Figs. 1–4 cover the scales of the transition and exhibit persistent randomness which is rooted in fluctuations of quantum nature, at the level of the monomers’ components and the attachment processes. At the other end of the universal scale galaxies (and their stars) exhibit similar fractal structures dominated by randomness [23], once again evidence of the existence of fundamental natural mechanisms at all scales. For example, reducing the spheres (representing electrons) to points in Fig. 5 one can obtain qualitatively the image of a galaxy of stars, or a section of our own galaxy.
6. Concluding remarks There is an undeniable relation between fractal and quantum physics like the aggregation of atoms and quantum paths and interactions. The isomorphism of the mathematics describing physical phenomena from widely different fields, like for example, the application of BCS techniques of superconductivity to the realm of nuclear physics or the Brownian motion–Schr€ odinger equations similarity, may indicate the existence of a possible unifying basis for fractality
822
R.J. Slobodrian / Chaos, Solitons and Fractals 14 (2002) 817–822
and quantization, to be much more fundamental and far reaching. There is a vast difference in scale between subatomic systems and the typical fractal aggregates, but fractal properties have been shown to be present even in quantum systems having permanent internal motion for their ‘‘instantaneous’’ configurations. This indicates that time dependent quantized systems persist in having fractal properties down to very small space-time scales. Continuity and smoothness at macroscopic scales of media and force fields is a misleading illusion. It comes about due to the statistics of large number of events and may be a reasonable approximation, but nature is basically discrete and stochastic. The starting point of quantum mechanics with differential equations, continuity conditions and Hilbert spaces may have to be replaced by an approach based on discretness and randomness, for space-time, particles, charges and force fields alike.
Acknowledgements The points discussed in this paper correspond to work and experience gathered during a period of some 12 years on fractals, mostly based on experimental evidence. Many scientists and technicians have contributed to the success of this research and it would be too long to cite them all here. However it is my pleasure to thank warmly Dr. Claude Rioux who has carried out much of the work in low gravity for his unfailing dedication and competence throughout all these years. The support of the Canadian Space Agency was essential to the experimental programme. Universite Laval has also contributed significantly with infrastructure and technical support.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
Mandelbrot B. Les objets fractals, forme, hasard et dimension. 3rd ed. Paris: Flammarion; 1989. Feder J. Fractals. New York: Plenum Press; 1988. Edgar GA, editor. Classics on fractals. Reading, MA: Addison Wesley; 1993. Kr€ oger H. Phys Rep 2000;323:81, and Refs. therein. This work reviews in depth fractal geometry relations with quantum physics. Slobodrian RJ. Phys Can 1995;51:245, and Refs. therein. Leclerc J-C, Rioux C, Slobodrian RJ. Chaos, Solitons & Fractals 2001;12:97. Gauthier M, Cossette M, Rioux C, Slobodrian RJ. Chaos, Solitons & Fractals 1993;3:687. Slobodrian RJ, Rioux C, Leclerc J-C. In: Sch€ umann B, editor. First International Symposium in Mycrogravity Research and Applications in Physical Sciences and Biotechnology, Vol. II. Noordwijk: ESTEC, ESA Publ. 2001; p. 779. Slobodrian RJ, Deladurantaye P, Gauthier M, Rioux C, Potvin L. Chaos, Solitons & Fractals 1996;7:25. Deladurantaye P, Rioux C, Slobodrian RJ. Chaos, Solitons & Fractals 1997;8:1693. Pierre Noyes H. SLAC-PUB-5189; 1990. Leclerc J-C. Proceedings of Spacebound 2000, May 2000, Available on CD-rom from Canadian Space Agency. Knight WD, Clemenger K, de Heer WA, Saunders WA, Chou MY, Cohen ML. Phys Rev Lett 1984;52:2141. Clemenger K. Phys Rev B 1985;32:1359. Brack M. Rev Mod Phys 1993;65:677. de Heer WA. Rev Mod Phys 1993;65:611. McCarthy IE. Introduction to nuclear theory. New York: Wiley; 1968. Baggot J. Perfect symmetry. UK: Oxford University Press; 1994. Cohen E. Comput Phys Commun 1992;70:441; Cohen E. Nucl Phys News 1992;5:20. Slobodrian RJ. Phys Rev C 1989;39:1052, and Refs. therein. Brack M, Bhaduri RK. Semiclassical physics. Reading, MA: Addison Wesley; 1997. Sommerfeld A. Phys Z 1911;12:1057, and Refs. therein. ec, Essiambre R.-J. Geometrie Fractale et Structure @ Grande echelle de l‘Univers, Memoire de ma^itrise. Universite Laval, Qub Canada; 1990.