An official website of the United States government
Classical electromagnetic laws such as Faraday's law are commonly interpreted within a continuum framework in which magnetic flux is treated as a statistically stable observable arising from large ensembles of magnetic dipoles. As magnetic systems approach the isolated nanoparticle limit, this continuum interpretation becomes progressively less robust because the magnetic response is increasingly governed by discrete localized sources rather than coarse-grained ensemble averaging. This work examines the implications of this continuum-to-discrete transition for the classical flux-based formulation of electromagnetic induction. While Maxwell's equations and the superposition principle remain fully valid at all length scales, the conventional interpretation of magnetic flux as a continuum observable loses operational robustness in the discrete-source regime. We show that electromagnetic induction is more naturally represented through the time-dependent vector potential generated directly by localized magnetic sources, providing a source-oriented description that remains fully consistent with classical electrodynamics. The implications for nanoscale magnetometry and isolated magnetic nanoparticles are discussed, identifying the transition from continuum field descriptions to discrete-source representations relevant to emerging nanoscale magnetic systems.
more »
« less
This content will become publicly available on June 1, 2027
Re-examining Magnetic Flux Density at the Nanoscale: A Statistical Perspective on Continuum and Discrete Regimes
Magnetic flux density (B) is traditionally interpreted as a continuous field whose flux lines form closed loops, as prescribed by Maxwell’s equations. This description is well justified in macroscopic systems, where large ensembles of magnetic dipoles produce statistically smooth fields through spatial averaging. At the nanoscale, however, where only a limited number of dipoles may contribute, the conditions underlying this continuum interpretation become less clear. Here, we reexamine the meaning of magnetic flux density from a classical-statistical perspective, focusing on finite ensembles and isolated magnetic particles. We show that as the number of contributing dipoles decreases, ensemble averaging becomes insufficient to support a statistically stable, coarse-grained field description, even though the underlying electromagnetic fields remain well defined and fully consistent with Maxwell’s equations. In this regime, magnetic flux density retains its formal definition, but its interpretation as a robust macroscopic observable becomes strongly dependent on fluctuations and specific dipole configurations. This framework introduces a quantitative criterion based on a critical particle number and provides a consistent description of the transition from ensemble-averaged magnetostatics to discrete dipole behavior. The results clarify the limits of continuum field interpretations at the nanoscale and offer a unified perspective for understanding isolated nanoparticles, small dipole ensembles, and the emergence of classical magnetic behavior from discrete microscopic sources.
more »
« less
- Award ID(s):
- 1953009
- PAR ID:
- 10696005
- Publisher / Repository:
- Springer nature
- Date Published:
- Journal Name:
- Journal of Superconductivity and Novel Magnetism
- Volume:
- 39
- Issue:
- 3
- ISSN:
- 1557-1939
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Abstract We establish the theoretical construct of an isolated magnetic nanoparticle, defined as a single magnetic particle free from inter-particle dipolar interactions. Within this framework, we demonstrate that the classical Maxwellian condition ∇·B = 0 breaks down at a critical nanoscale size (~ 25.7 nm), below which the condition loses physical relevance for isolated particles. At these dimensions, discrete dipole effects, flux quantization, and incomplete flux closure dominate, replacing the smooth field behavior of bulk systems. The requirement of a minimum cross-sectional area to sustain a single magnetic flux quantum provides a natural size threshold for magnetic stability at the nanoscale. This work establishes a conceptual and quantitative boundary between classical and quantum magnetism, offering new insights into nanoscale magnetic phenomena and guiding the design of novel magnetic nanostructures, spintronic elements, and quantum magnetic devices. Graphical Abstract Size-dependent magnetic flux in an isolated single particle:Large particle:Multiple continuous magnetic flux lines appear, consistent with classical behavior.Critical particle:Only a single flux line is supported; dipoles are insufficient for full flux closure.Sub-critical particle:No observable flux lines; intrinsic size effects suppress classical field manifestation. All particles are “ideal,” magnetically isolated, and interact only through internal dipoles.more » « less
-
When combined with nanostructured substrates, two-dimensional semiconductors can be engineered with strain to tailor light–matter interactions on the nanoscale. Recently, room-temperature nanoscale exciton localization with controllable wrinkling in 1L-WSe2 was achieved using arrays of gold nanocones. Here, the characterization of quantum dot-like states and single-photon emitters in the 1L-WSe2/nanocone system is reported. The nanocones induce a wide range of strains, and as a result, a diverse ensemble of narrowband, potential single-photon emitters is observed. The distribution of emitter energies reveals that most reside in two spectrally isolated bands, leaving a less populated intermediate band that is spectrally isolated from the ensembles. The spectral isolation is advantageous for high-purity quantum light emitters, and anti-bunched emission from one of these states is confirmed up to 25 K. Although the spatial distribution of strain is expected to influence the orientation of the transition dipoles of the emitters, multimodal emission polarization anisotropy and atomic force microscopy reveal that the macroscopic orientation of the wrinkles is not a good predictor of dipole orientation. Finally, the emission is found to change with thermal cycling from 4 to 290 K and back to 4 K, highlighting the need to control factors such as temperature-induced strain to enhance the robustness of this quantum emitter platform. The initial characterization here shows that controlled nanowrinkles of 1L-WSe2 generate quantum light in addition to uncovering potential challenges that need to be addressed for their adoption into quantum photonic technologies.more » « less
-
We consider electrostatic interactions in two classes of nanostructures embedded in a three dimensional space: (1) helical nanotubes, and (2) thin films with uniform bending (i.e., constant mean curvature). Starting from the atomic scale with a discrete distribution of dipoles, we obtain the continuum limit of the electrostatic energy; the continuum energy depends on the geometric parameters that define the nanostructure, such as the pitch and twist of the helical nanotubes and the curvature of the thin film. We find that the limiting energy is local in nature. This can be rationalized by noticing that the decay of the dipole kernel is sufficiently fast when the lattice sums run over one and two dimensions, and is also consistent with prior work on dimension reduction of continuum micromagnetic bodies to the thin film limit. However, an interesting contrast between the discrete-to-continuum approach and the continuum dimension reduction approaches is that the limit energy in the latter depends only on the normal component of the dipole field, whereas in the discrete-to-continuum approach, both tangential and normal components of the dipole field contribute to the limit energy.more » « less
-
Abstract The effects of strong magnetic fields on the deconfinement phase transition expected to take place in the interior of massive neutron stars are studied in detail for the first time. For hadronic matter, the very general density-dependent relativistic mean field model is employed, while the simple, but effective vector-enhanced bag model is used to study quark matter. Magnetic-field effects are incorporated into the matter equation of state and in the general-relativity solutions, which also satisfy Maxwell’s equations. We find that for large values of magnetic dipole moment, the maximum mass, canonical mass radius, and dimensionless tidal deformability obtained for stars using spherically symmetric Tolman–Oppenheimer–Volkoff (TOV) equations and axisymmetric solutions attained through the LORENE library differ considerably. The deviations depend on the stiffness of the equation of state and on the star mass being analyzed. This points to the fact that, unlike what was assumed previously in the literature, magnetic field thresholds for the approximation of isotropic stars and the acceptable use of TOV equations depend on the matter composition and interactions.more » « less
