Filling Space With Polydisperse Spheres In A Non-Apollonian Way

Completing a Swiss-cheese theorem of Lieb and Lebowitz (LL), we prove that any population of spheres with power-law radius distribution alpha 1/r(df+1) can completely fill 3D Euclidean space if the exponent is such that 2.8 <= d(f) < 3. This sufficient condition extends considerably the known part of the ensemble of space-filling populations of polydisperse spheres. The self-similar spatial arrangement of the polydisperse spheres related to the theorem is discussed using a numerical example with d(f) = 2.875. By calculating the small-angle scattering structure factor of the resulting packing, we found it to present several crystalline peaks indicating some regularity. This is significantly different from the featureless structure factor of an Apollonian packing which represents total disorder. We thereby argue that the LL algorithm for filling space with spheres is fundamentally different from Apollonian constructions.