Searching for the nanoscopic–macroscopic boundary

• A novel quantum-Heisenberg variational energy method is implemented.
• The asymptotic behavior toward the thermodynamic limit is explored.
• An important dependence of the nano-bulk boundary on the geometry is found.
• And also an important dependence on the crystalline lattice.
• We obtain a very good agreement with experimental evidence with differences <2%<2%.

Several studies have focused on the size-dependent properties of elements, looking for a unique definition of the nanoscopic–macroscopic boundary. By using a novel approach consisting of an energy variational method combined with a quantum Heisenberg model, here we address the size at which the ordering temperature of a magnetic nanoparticle reaches its bulk value. We consider samples with sizes in the range 1–500 nm, as well as several geometries and crystalline lattices and observe that, contrarily to what is commonly argued, the nanoscopic-microscopic boundary depends on both factors: shape and crystalline structure. This suggests that the surface-to-volume ratio is not the unique parameter that defines the behavior of a nanometric sample whenever its size increases reaching the bulk dimension. Comparisons reveal very good agreement with experimental evidence with differences less than 2%. Our results have broad implications for practical issues in measurements on systems at the nanometric scale.