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Approximate explicit formulas for Stoner-Wohlfarth hysteresis loops

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Pith's one-line read Approximate elementary formulas reproduce Stoner–Wohlfarth hysteresis loops for single particles and random ensembles across the full field range.

desk verdict Clean, usable elementary approximations to SW loops that actually work; soft spots are openly quantified and do not sink the claim. read the letter →

arxiv 2607.09881 v1 pith:LGYFFXMD submitted 2026-07-10 cond-mat.mtrl-sci

Savin Vladimir P. , Koksharov Yury A This is my paper · ORCID
classification cond-mat.mtrl-sci
keywords Stoner–Wohlfarthmodelhysteresisloopmagneticnanoparticlessingledomainanalyticalformulasastroideasyaxisrandomanisotropy
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper derives closed-form approximate expressions for the magnetization curves of Stoner–Wohlfarth particles. It covers both a particle whose easy axis is fixed relative to the applied field and an ensemble whose axes are randomly oriented. The authors start from a simple physical picture: at low field the moment stays near the easy axis, while at high field it stays near the field direction. Those small-angle expansions, joined at carefully chosen intermediate field values, produce piecewise formulas built only from elementary functions. The resulting expressions match numerical solutions of the classic model to within a few percent over most of the field range, the largest (still modest) discrepancy occurring near the sharp slope change of the averaged loop. The formulas therefore give a practical, integration-free alternative for calculating SW hysteresis without numerical root-finding or angular averaging.

What carries the argument

The low-field expansions for the equilibrium angle θ* (Eqs. 8–9) and the high-field expansion (Eq. 12), joined at hand-chosen matching fields (h0 = ±1.5 for fixed-angle loops, h0 = ±0.75 for the averaged loop) that minimize global deviation from the numerical solution.

What would settle it

Direct numerical comparison of the piecewise formulas against the exact energy-minimization solution of the Stoner–Wohlfarth equations over a dense grid of field values and easy-axis angles; if the absolute magnetization error systematically exceeds ~0.05 outside the immediate vicinity of switching, or if the averaged-loop coercivity error grows beyond a few percent, the claim fails.

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Extended reading notes

Core claim

Piecewise elementary formulas, obtained by matching low-field (moment near easy axis) and high-field (moment near applied field) expansions at optimized intermediate points, reproduce the full Stoner–Wohlfarth hysteresis loops for both fixed easy-axis orientation and randomly oriented ensembles, with absolute magnetization error remaining below roughly 0.05 except near the switching field.

Load-bearing premise

The small-angle expansions that keep the magnetic moment near the easy axis remain accurate enough even near the Stoner–Wohlfarth switching curve, where the reduced field is no longer small.

Editorial extensions

If this is right

  • Hysteresis loops for fixed-angle and randomly oriented SW particles can be evaluated with only elementary functions, without root-finding or angular integration.
  • Initial susceptibilities of both single particles and random ensembles recover the known analytic limits of the model.
  • The same piecewise construction supplies an explicit initial-magnetization curve for the random ensemble.
  • The formulas remain usable near the switching field, where previous low-field approximations break down.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The matching-field choice could be automated by a simple least-squares or minimax criterion, turning the present hand-tuned intervals into a fully algorithmic construction.
  • Because the formulas are elementary, they can be embedded directly in larger micromagnetic or finite-element codes that currently call numerical SW solvers at every mesh point.
  • The same low-/high-field matching strategy may extend to other uniaxial anisotropy models whose energy landscapes share a similar two-regime structure.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 5 minor

Summary. The paper derives approximate explicit piecewise formulas for Stoner–Wohlfarth hysteresis loops, both for a single particle at fixed easy-axis angle θ_H (Eqs. 16–20) and for an ensemble with randomly oriented anisotropy axes (Eqs. 50–55). The derivations rest on low-field small-angle expansions of the equilibrium condition (moment near the easy axis, Eqs. 8–9) and a high-field expansion (moment near the field direction, Eq. 12), with matching abscissae chosen to minimize global error against numerical solutions of the same model. Absolute deviations remain ≲0.05 over most of the field range, with the largest (still described as reasonably small) deviation near h_0≈0.5 for the averaged loop.

Significance. If the stated accuracy holds, the formulas supply a practical, elementary-function alternative to numerical root-finding and angular integration for the classic SW model. They rest on a transparent physical picture rather than pure least-squares parametrization, recover known limits (initial susceptibilities, approach-to-saturation law), and cover both fixed-orientation and random-ensemble cases. The internal validation against exact numerical solutions of the SW equations, together with the explicit inverse-astroid formulas in Appendix A, makes the results immediately usable for modeling nanoparticles, thin films, and polycrystalline magnets without statistical sampling.

minor comments (5)
  1. In §3.1–3.2 the small-angle approximations are introduced without an a-priori error bound; a short remark quantifying the maximum |θ′| or |θ*| on the astroid (already plotted in Fig. 5) would help the reader judge the domain of validity before the numerical checks of §6.
  2. The matching fields |h_0|=1.5 (fixed-θ_H) and |h_0|=0.75 (ensemble) are stated to be chosen for best fit, yet no sensitivity analysis is given. A single sentence or inset showing how the max |δ| changes if the join is moved by ±0.1 would strengthen the claim that the choice is robust.
  3. Figure 2 caption and the surrounding text refer to “dotted curves” for the analytic formulas, but the line styles are hard to distinguish in grayscale; adding a legend or using dashed/solid consistently would improve readability.
  4. Eq. (23) for the initial curve at fixed θ_H is left in a rather cumbersome product-of-sines form; a brief expansion for small h_0 would make the connection to the known χ_max = M_0²/(2K_a) more transparent.
  5. Typographical: in the abstract and highlights the arXiv identifier appears as 2607.09881 while the header uses the same; consistency with the final journal citation style should be checked at proof stage.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: approximations are asymptotic expansions of the SW equilibrium equation, assembled piecewise and validated against independent numerical solutions of the same model.

full rationale

The derivation chain begins from the exact SW reduced energy (Eq. 1) and equilibrium condition (Eq. 3). Low-field (sin heta'≈ heta', cos heta'≈1) and high-field expansions then produce the explicit heta* approximations (Eqs. 8, 9, 12) that are substituted into the projection m_H0/m0=cos( heta*- heta_H) and, for the ensemble, into the orientation integral (Eq. 24). The resulting piecewise elementary expressions (fixed- heta_H: Eqs. 16–20; random: Eqs. 50–55) are therefore approximate solutions of the model itself. Matching abscissae (h0=±1.5, ±0.75) and the order of the logarithm expansion are chosen post-hoc to minimize absolute deviation from numerical solutions of the identical SW equations; this is ordinary engineering of an approximation, not a fit of free parameters that are later re-labeled as independent predictions. No self-citation supplies a load-bearing uniqueness claim, no ansatz is smuggled via prior work of the authors, and the formulas are not definitional rearrangements of their inputs. The paper is self-contained against its own numerical benchmarks; circularity score is therefore zero.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The paper works entirely inside the classical Stoner–Wohlfarth energy functional. The only free choices are the numerical abscissae at which low- and high-field pieces are joined; all other ingredients are either standard SW assumptions or elementary small-angle expansions. No new physical entities are postulated.

free parameters (3)
  • matching field |h0|=1.5 (fixed-angle loops)
    Chosen by hand so that the low-field cosine formulas and the high-field square-root formula join with minimal discontinuity and best visual agreement with numerical data (Section 4).
  • matching fields |h0|=0.75 and 0.5 (ensemble loops)
    Region boundaries selected to keep absolute error ≲0.05 while preserving simple analytic expressions (Section 5.4 and Fig. 7).
  • third-order truncation of ln(1-h0)
    Expansion order chosen because it reduces the coercivity error relative to the unexpanded or second-order forms (Section 5.1).
assumptions (3)
  • domain assumption Stoner–Wohlfarth reduced energy w = (1/2)sin^{2}θ − h0 cos(θ−θH) and the associated equilibrium/stability conditions (Eqs. 1–4)
    Standard continuum model of a uniaxial single-domain particle; taken as given throughout.
  • ad hoc to paper Small-angle approximations sin θ′≈θ′, cos θ′≈1 (and likewise for θ″) when the moment is near the easy axis or the field direction
    Core working hypothesis of Sections 3.1–3.3; validity near the astroid is checked a posteriori rather than proved a priori.
  • domain assumption Uniform (isotropic) distribution of anisotropy axes for the ensemble average (Eq. 24)
    Classical idealization of a non-interacting polycrystalline powder; used without further justification.

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Cite this review

Pith. "Pith review of Approximate explicit formulas for Stoner-Wohlfarth hysteresis loops." pith.science (2026). https://pith.science/paper/LGYFFXMD

@misc{pith2026260709881,
  author       = {Pith},
  title        = {Pith review of: Approximate explicit formulas for Stoner-Wohlfarth hysteresis loops},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LGYFFXMD}},
  note         = {Machine review of arXiv:2607.09881}
}
read the original abstract

Approximate explicit formulas for the hysteresis loops in the Stoner-Wohlfarth model are derived. We consider the hysteresis loops both for a single particle with a fixed easy-axis direction and for an ensemble of particles with randomly oriented anisotropy axes. The physical assumption used to derive the formulas is that the particle magnetic moment lies in the vicinity of the easy axis or the external field direction, at low and high fields, respectively. Surprisingly, the low-field formula is approximately valid even near the Stoner-Wohlfarth astroid, where the reduced magnetic field h0 is not very small. The general piecewise formula is obtained by an appropriate matching of the functions defined on different intervals of the magnetic field, which are chosen to maximize the formula accuracy. For the averaged hysteresis loop, the maximal, but reasonably small, deviation of our formula from numerically calculated magnetization occurs at h0 = 0.5, which corresponds to the sharp change in magnetization slope.

Figures

Figures reproduced from arXiv: 2607.09881 by the authors.

Figure 1
Figure 1. Schematic diagram of a Stoner–Wohlfarth (SW) particle with uniaxial anisotropy along [PITH_FULL_IMAGE:figures/full_fig_p026_1.png] view at source ↗
Figure 2
Figure 2. Magnetization curves for a SW particle for fixed angles [PITH_FULL_IMAGE:figures/full_fig_p027_2.png] view at source ↗
Figure 3
Figure 3. Reduced average magnetic moment ⟨mH0 ⟩/m0 as a function of h0. Solid curves show the results of numerical calculations using Eqs. (3)–(5) and (24). Dotted curves are plotted using the following analytical equations: Eqs. (50) and (51) in Region I; Eqs. (52) and (53) in Region II; Eqs. (54) and (55) in Region III. 26 [PITH_FULL_IMAGE:figures/full_fig_p028_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: The dependence of h0,⊥ on h0,∥. This scheme illustrates the partition of the integration interval. The thick dotted lines represent the astroid; circular arcs correspond to areas with a constant applied field magnitude (h0 = |h0| = const). The field direction is specif…
Figure 5
Figure 5. Figure 5: (a) The angles θ a and θ b versus θH. The dotted line corresponds to π − θ a , which rep￾resents the deviation of the magnetic moment from the anisotropy axis just before switching. The angle θ a was calculated analytically using Eq. (61). The numerical solution for θ …
Figure 6
Figure 6. Figure 6: (a) Absolute deviation |δ| between the analytical solution (Eqs. (16)–(20)) and the nu￾merical solution for the ascending branches of hysteresis loops versus h0 for fixed θH (numbers 1–5 correspond to the magnetization curves in [PITH_FULL_IMAGE:figures/full_fig_p031_6.png]
Figure 7
Figure 7. Figure 7: Absolute deviation |δ| between the approximate analytical solution and the numerical solution for the ascending branch of the averaged hysteresis loop (see [PITH_FULL_IMAGE:figures/full_fig_p032_7.png]

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