Fractal dimension as a scaling law for nuclear quantum effects: a neutron Compton scattering study on carbon allotropes

In this work, we tackle the problem of the sensitivity of neutron Compton scattering, measured through the widths of nuclear momentum distributions, to the degree of complexity and ordering of the structural motif characterising the surrounding environment felt by a particular nucleus (carbon). In doing so, we replace the usual concept of the bond strength categorised in terms of its thermodynamical or electronic properties with a novel observable inspired by the language of mathematical topology, the Hausdorff-Besicovitch fractal dimension. We derive a relatively straightforward connection between the fractal dimension of a given system under consideration and the nuclear kinetic energy. To achieve this, we modify the concept of the energy equipartition theorem for solid-state systems composed of carbon atoms where the atom-ordering topology does not follow a simple two or three-dimensional order, but rather atoms are placed along curves in space that have an intermediate dimension related to the varying amounts of information they contain. A series of results from past neutron Compton scattering studies, as well as new results on Buckminsterfullerene (C-60), correlate with the topological measures of surface roughness and bending, as categorised quantitatively by the fractal dimension of the system. Namely, for the same formal chemical binding motif (sp(2) C) and with decreasing system dimensionality from nearly 3 towards 1, the quantum nature of the system becomes more pronounced. The simple scaling law developed in this work allows for relatively simple assessment of the nuclear "quantumness" of a given system with potentially important ramifications in the ab initio modelling of nuclear quantum effects in condensed matter.