Representations based on the concepts of the theory of fuzzy sets are suggested to apply to a wide variety of problems in physics. A grade of membership is associated to each element in a set, which is a measure of its distance to a prototype. Fuzzy representations are thus adequate for dealing with situations where the belongingness of an object or a phenomenon to a class is uncertain, or to situations where the classes have no exact definition. An explicit relation is shown between fuzzy representation and dimensionality. Unambiguous definitions of the degree of fuzziness, cluster overlap, and isolated points are given on the basis of an anisotropic grade of membership function. The example is treated of collision cascades generated by xenon atoms incident on a polycrystalline gold surface with energies ranging from 20 keV to 1 MeV. The cascades are simulated in the binary collision approximation with the Marlowe computer code. They are shown to germinate from simultaneously growing collisions clusters. The displacement cascades are found to be only partially space filling. This is emphasized on the basis of their fuzzy geometrical characteristics, without need of any assumption concerning self-similarity. Their possible overlap and lumping are identified on the basis of the grade of membership of each vacated lattice site to each cluster. The final cluster pattern of the vacancy distributions is shown to depend on the degree of fuzziness. The sensitivity of several properties of vacancy clusters on the degree of fuzziness is discussed. This sensitivity is suggested to be a consequence of their granular structure. Consequently, their experimental characterization may be influenced by the resolution of the observation method. Fuzzy analysis is suggested as a tool to establish the relation between measures at different scales of the same phenomenon.

• Received 12 August 1988

DOI:https://doi.org/10.1103/PhysRevA.39.2817