Factors influencing the density of nanoparticles include the intrinsic density of the element or compound that forms the nanoparticle, surface properties, pore structure, and agglomeration within suspensions or aggregates. In this case, the first factor is the main one; therefore, the methods for quickly assessing changes in nanoparticle density as a function of particle size, which we considered in this paper, are relevant. An estimate of the actual density of a substance is proposed depending on the number of atoms in a particle (nanoparticle). It is found that density oscillations are observed in the dependence of the nanoparticle density on its radius. Calculations are carried out for C (diamond, graphite, graphene), Fe, Si, and ZrO2. When density is considered as a function of the number of atoms, oscillations are observed, which vanish at a content of 104 atoms, after which the density approaches values similar to those of bulk materials. The values of the radius of metal microparticles correspond to the values obtained earlier by other authors when explaining the deviation of the specific heat capacity of nanoparticles from the macroscopic heat capacity, based on solutions of the Schrödinger equation. The threshold values of mass and radius are unique for each substance and are significantly smaller than the Planck mass.
Arsent’ev et al. (2025) studied this question.