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. 2018 Jul 24;8:11166. doi: 10.1038/s41598-018-29501-8

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© The Author(s) 2018

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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The reduced magnetisation as a function of temperature for a field-cooled (FC, solid lines) and zero-field-cooled (ZFC, dashed lines) random assembly of particles, calculated three different ways. Panel (a) shows the reduced magnetisation calculated using Eqs (1 and 2). Fe₂O₃ particles all with 5 nm radius are considered (M_s = 281 kA/m and K = 16 kJ/m³) with a weak applied field H = 10 Oe (796 A/m). These parameters are chosen to match ref.¹³. In panel (b), the magnetisation curves are re-drawn using a log-normal distribution of particle sizes with dispersion σ = 0.02 using Eqs (5 and 6) and the sharp peak is smoothed out. In panel (c) our weighted-probability expressions, to be developed later in the text in Eqs (10 and 11), are drawn for particles of a single size and show a similar shape to the plots in panel (b) where a size distribution is assumed. The measurement time is taken to be ln (τ_m/τ₀) = 25. The vertical line in panels (b) and (c) indicates the mode blocking temperature, to the left of the ZFC peak.