Inverse-square law interaction at the nanoscale

The interactions between the apples of the tree and the earth (near-field) and the ones governing the planetary systems around some of those stars (far-field) could be described using an inverse-square law.

Any physical law in which the magnitude of a physical quantity is proportional to the reciprocal of the square of the distance (1/r2) from the source of that property is known as an inverse-square law. Newton’s law of gravitation and Coulomb’s law are both examples.

When Newton proposed his law of gravitation in his Philosophiæ Naturalis Principia Mathematica (1687), which was a mathematical description of the movement of planets and other objects following an inverse square law, he was repeatedly asked about the origin of that spooky (as Einstein famously described it) action at a distance, the concept that an object can be moved, changed, or otherwise affected without being physically touched (as in mechanical contact) by another object. He answered in an essay, “General Scholium”, which was appended to the second (1713) edition of the Principia.:

I have not as yet been able to discover the reason for these properties of gravity from phenomena, and I do not feign hypotheses. For whatever is not deduced from the phenomena must be called a hypothesis; and hypotheses, whether metaphysical or physical, or based on occult qualities, or mechanical, have no place in experimental philosophy. In this philosophy particular propositions are inferred from the phenomena, and afterwards rendered general by induction. 1

But not everyone was happy with this “hypotheses non fingo”. Nicolas Fatio de Duillier in 1690 and later Georges-Louis Le Sage in 1748 proposed a kinetic theory of gravity to explain the inverse-square law. The theory proposed a mechanical explanation for Newton’s gravitational force in terms of streams of tiny unseen particles (which Le Sage called ultra-mundane corpuscles) impacting all material objects from all directions. According to this model, any two material bodies partially shield each other from the impinging corpuscles, resulting in a net imbalance in the pressure exerted by the impact of corpuscles on the bodies, tending to drive the bodies together. This theory for gravity never gained traction, though. Still, Le Sage is considered one of the foreparents of the kinetic theory, so successful in the atomic scale.

Fatio de Duillier–LeSage corpuscular theory was generalized to electromagnetic waves by Lorentz who proposed that, due to their mutual shadowing, two absorbing particles in an isotropic radiation field experience an attractive force that follows a gravity-like inverse square distance law. Similar interactions were later introduced by Spitzer (1941) and Gamow (1949), who called it “mock gravity”, in the context of galaxy formation, but their actual relevance in cosmology has never been unambiguously established.

Recent results have demonstrated that the interaction force between non-absorbing dielectric particles and atoms in a quasi-monochromatic isotropic random light field always presents an oscillatory behavior for distances larger than the light wavelength. Now a team of researchers shows 2 that, under specific resonant conditions, these forces become non-oscillating, 1/r2, attractive forces. In other words, at the resonant condition, the interaction forces follow a long-range mock gravity law.

The interaction between two objects is usually defined to be long ranged if the force decays with their distance apart, r, as a power law ∼1/rn+1 , with n smaller than the spatial dimension of the system. Gravity is a typical example of a long-range attractive force in three dimensions while the interaction between electric or magnetic dipoles (n = 3) is borderline in between short and long range attraction.

In contrast, dispersion forces between non-polar, neutral molecules and particles are short range: at close distances the Coulomb interaction between the fluctuating electric dipole moments leads to the van der Waals–London interaction energy proportional to 1/r6. However, when r is larger than a characteristic resonance wavelength, the interaction energy varies as 1/r7.

For small separation distances, gravity-like interactions were first predicted by quantum electrodynamics calculations for atoms and molecules and later by a classical approach on Rayleigh nanoparticles leading to analogous results. In these studies, the imaginary part of the polarizability was either not included in the calculations or was taken into account but considered negligible; i.e., radiation pressure effects were neglected. However, as the researchers show now, these effects dominate the near-field interactions of non-absorbing particles, leading to a different interaction law.

The authors analyzed the interaction forces between identical resonant nanoparticles, in free space or homogeneous media, illuminated by a quasi-monochromatic isotropic random light field. They found that, in contrast with atoms or dielectric particles, the interaction force between plasmonic nanoparticles, whose extinction cross section is dominated by absorption, can follow a true attractive inverse square law all the way from near to far-field separation distances.

Author: César Tomé López is a science writer and the editor of Mapping Ignorance

Disclaimer: Parts of this article may have been copied verbatim or almost verbatim from the referenced research paper.


  1. Isaac Newton (1726). Philosophiae Naturalis Principia Mathematica, General Scholium. Third edition, page 943 of I. Bernard Cohen and Anne Whitman’s 1999 translation, University of California Press ISBN 0-520-08817-4,
  2. J. Luis-Hita, M. I. Marqués, R. Delgado-Buscalioni, N. de Sousa, L. S. Froufe-Pérez, F. Scheffold, and J. J. Sáenz (2019) Light Induced Inverse-Square Law Interactions between Nanoparticles: “Mock Gravity” at the Nanoscale Phys Rev. Lett. Doi: 10.1103/PhysRevLett.123.143201

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