Fractal dimensions of jammed packings with power-law particle size distributions in two and three dimensions

Static structure factors are computed for large-scale, mechanically stable, jammed packings of frictionless spheres (three dimensions) and disks (two dimensions) with broad, power-law size dispersity characterized by the exponent $\ensuremath{-}\ensuremath{\beta}$. The static structure factor exhibits diverging power-law behavior for small wave numbers, allowing us to identify a structural fractal dimension ${d}_{f}$. In three dimensions, ${d}_{f}\ensuremath{\approx}2.0$ for $2.5\ensuremath{\le}\ensuremath{\beta}\ensuremath{\le}3.8$, such that each of the structure factors can be collapsed onto a universal curve. In two dimensions, we instead find $1.0\ensuremath{\lesssim}{d}_{f}\ensuremath{\lesssim}1.34$ for $2.1\ensuremath{\le}\ensuremath{\beta}\ensuremath{\le}2.9$. Furthermore, we show that the fractal behavior persists when rattler particles are removed, indicating that the long-wavelength structural properties of the packings are controlled by the large particle backbone conferring mechanical rigidity to the system. A numerical scheme for computing structure factors for triclinic unit cells is presented and employed to analyze the jammed packings.