Fractal geometry of collision cascades
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Michael Nastasi Materials Science and Technology Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (Received 1 April 1988; accepted 13 September 1988)
The fractal nature of self-ion collision cascades is first described using an inverse power potential and then by the more realistic potential of Biersack-Ziegler. Based on the model of Cheng et al. and TRIM Monte Carlo simulations, the average cascade fractal dimension is a function of both atomic mass and initial energy. The instantaneous fractal dimension increases as the cascade evolves. A critical energy Ec for producing a dense subcascade is derived and it is shown that Ec agrees well with the onset energy for constant damage efficiency. I. INTRODUCTION Fractal geometry was introduced and developed by B. Mandelbrot,1 and has found applications in many different domains, including materials science. Cheng et al.2 applied fractal geometry to a displacement cascade and developed a description of some of the general properties of a self-ion collision cascade. Winterbon,3 using a different approach, has also described fractal aspects of collision cascades. Using an analogous approach to that of Cheng et al.,1 in this paper, we describe a self-ion cascade in terms of fractal geometry, basing it upon the BiersackZiegler potential, 4 and then compare the results with Monte Carlo simulations. The first part of the work reviews some basic concepts of fractal geometry, and emphasizes the properties of a displacement cascade which are related to fractal geometry. The second part describes a fractal geometry analysis of a cascade with the inverse power potential and the BiersackZiegler potential. Using the Biersack-Ziegler potential allows the formulation of a characteristic fractal dimension which is related in the third part to the results of Monte Carlo simulations. The comparison then leads to the discussion of the general concepts of the displacement spike. The final part deals with a comparison of the present results with published experimental and molecular dynamics simulation data. II. SOME FRACTAL GEOMETRY CONCEPTS Two properties of fractals are central to this discussion: the fractal dimension and self similarity. These two principles are succinctly described in the following and their relationship to the present problem introduced. A. Fractal dimension A classical example of the application of the fractal dimension concepts is the measurement of the length of coastlines. It has been known for a long time that if someone tries to measure the length of a coastline by Permanent address: Commissariat a l'Energie Atomique, Centre de Valduc, 21120 Is sur Tille, France. a|
adding "unit segments" of a given length, the results obtained tend to infinity as the segment length used for the measurement tends to zero. This can be easily explained since the use of a smaller unit allows the inclusion of more details, whose lengths are added to the final result. It is then impossible to determine the real length of a coastline by simply adding unit segment
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