arxiv: 2604.12764 · v1 · submitted 2026-04-14 · ❄️ cond-mat.mtrl-sci · physics.class-ph

Recognition: unknown

Exact demagnetisation field for periodic one-dimensional array of rectangular prisms

Frederik Laust Durhuus , Andrea Roberto Insinga , Rasmus Bj{\o}rk

Authors on Pith no claims yet

Pith reviewed 2026-05-10 14:49 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci physics.class-ph

keywords demagnetizing fieldrectangular prismsperiodic arrayanalytical solutionmicromagnetic simulationpoint dipolesmagnetostatic field

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The pith

The paper derives an analytical solution for the demagnetizing field of an infinite periodic array of rectangular prisms that is exact on the axis for thin prisms.

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

The authors derive a closed-form analytical solution for the magnetic field produced by an infinite chain of uniformly magnetized rectangular prisms placed end to end. This solution is exact along the central axis in the limit where each prism becomes infinitesimally thin. The result improves on existing approximations used in micromagnetic modeling of periodic structures by providing a rapidly converging series that can be evaluated directly. It also yields an analogous expression for an array of point dipoles. The work matters because accurate demagnetizing fields are essential for reliable simulations of magnetic materials and devices.

Core claim

We derive an analytical solution for the field from a periodically repeating infinite array of prisms aligned end-to-end, which becomes exact on the center axis in the limit of infinitesimally thin prisms. Using the same method we derive the on-axis field for a one-dimensional array of point dipoles. We validate the obtained results numerically and furthermore compare with the common macrogeometry approach and more recent uniform magnetisation method, demonstrating an excellent convergence rate for the novel method.

What carries the argument

Summation of the known single-prism demagnetizing field over the infinite periodic lattice, evaluated exactly on the central axis after taking the thin-prism limit.

If this is right

The derived series converges faster than the macrogeometry approximation and the uniform magnetization method.
Numerical checks confirm that the analytical expressions match direct computations of the field.
The same summation technique produces an exact on-axis field for a periodic array of point dipoles.
The expressions supply a practical tool for computing demagnetizing fields in one-dimensional periodic magnetic structures.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The on-axis exact result can serve as a benchmark for testing numerical codes that model periodic magnetic chains.
The summation approach may be adapted to compute fields at off-axis locations or for finite numbers of prisms if the single-prism field remains known.
This class of exact thin-limit results could guide the design of more efficient micromagnetic models for elongated magnetic elements such as nanowires.

Load-bearing premise

The derivation assumes uniform magnetization inside each prism and becomes exact only on the central axis when the prism thickness approaches zero.

What would settle it

A direct numerical integration or finite-element calculation of the field on the axis for a long but finite chain of very thin prisms should agree with the analytical series to within numerical error; systematic disagreement would falsify the exactness claim.

Figures

Figures reproduced from arXiv: 2604.12764 by Andrea Roberto Insinga, Frederik Laust Durhuus, Rasmus Bj{\o}rk.

**Figure 2.** Figure 2: FIG. 2: Illustration of an array of identical point [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3: Illustration of special case where the analytical [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4: Illustration of uniform magnetisation [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5: Convergence tests [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗

read the original abstract

The magnetic field from a uniformly magnetised, rectangular prism is known exactly, which is the basis for a large number of micromagnetic simulations. Here we derive an analytical solution for the field from a periodically repeating infinite array of prisms aligned end-to-end, which becomes exact on the center axis in the limit of infinitesimally thin prisms. Using the same method we derive the on-axis field for a one-dimensional array of point dipoles. We validate the obtained results numerically and furthermore compare with the common macrogeometry approach and more recent uniform magnetisation method, demonstrating an excellent convergence rate for the novel method.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives a clean closed-form expression for the demagnetizing field along the axis of a periodic 1D array of rectangular prisms, exact in the thin-prism limit.

read the letter

The main thing here is a new analytical expression for the demagnetizing field of an infinite periodic array of rectangular prisms lined up end to end. It reduces to an exact closed form on the central axis when the prisms are made infinitesimally thin. They build this by taking the known exact solution for one prism and summing the contributions from all the periodic images analytically. That gives a clean formula without needing to truncate the sum numerically. They also do the same for point dipoles. The numerical checks look solid: the new expression converges quickly and matches the macrogeometry and uniform magnetization approaches in the appropriate limits. The derivation is transparent and rests on standard assumptions like uniform magnetization inside each prism. The main limitation is that exactness is restricted to the axis and the thin-prism limit; for finite thickness or off-axis points it becomes an approximation, though still useful. No free parameters or circular fitting, which is good. This is the kind of incremental but practical advance that people in micromagnetic modeling will appreciate, especially for periodic nanostructures. The math checks out on its own terms and the validation is straightforward. I would bring this to a reading group if the group works on magnetic materials or numerical methods in magnetism. It is worth sending to peer review because the result is new, the evidence supports it, and it fills a small but real gap in the toolkit for these calculations.

Referee Report

0 major / 3 minor

Summary. The paper derives an analytical expression for the demagnetizing field of an infinite periodic one-dimensional array of uniformly magnetized rectangular prisms aligned end-to-end. The central result is obtained by superposing the known exact single-prism field and reduces to a closed-form expression that is exact on the center axis in the limit of vanishing prism thickness. The same approach yields the on-axis field for a 1D lattice of point dipoles. Results are validated by direct numerical summation and compared against the macrogeometry approximation and the uniform-magnetization method, showing rapid convergence.

Significance. If the derivation holds, the work supplies a parameter-free, analytically closed expression for a common geometry in micromagnetic modeling. This is a direct extension of the established single-prism solution and therefore inherits its rigor while removing the need for numerical lattice sums or fitted demagnetizing factors in the thin-prism limit. The explicit point-dipole result and the documented numerical checks against two independent methods constitute reproducible, falsifiable content that can be immediately adopted in simulation codes.

minor comments (3)

[Abstract] The abstract states that the solution 'becomes exact on the center axis in the limit of infinitesimally thin prisms' but does not restate the uniform-magnetization assumption that underlies the starting single-prism field; a single clarifying clause would prevent misreading.
[Derivation section] Equation numbers for the infinite periodic sum (the superposition step) and for the final closed-form on-axis expression are not cross-referenced in the text; adding these references would improve traceability.
[Numerical validation] The convergence plots compare the new method to macrogeometry and uniform-magnetization results, but the figure captions do not specify the exact error norm (e.g., L2 or pointwise) or the number of terms retained in the reference sums.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The report contains no major comments or criticisms requiring a point-by-point response.

Circularity Check

0 steps flagged

No significant circularity; derivation is an analytic superposition of an externally known single-prism result

full rationale

The paper starts from the established exact demagnetizing field of an isolated uniformly magnetized rectangular prism (a result external to this work) and performs an analytic summation over a one-dimensional periodic lattice. The central closed-form expression on the axis in the thin-prism limit follows directly from this summation under the uniform-magnetization assumption; no parameters are fitted to the target data, no self-citation supplies a load-bearing uniqueness theorem or ansatz, and the result is not equivalent to its inputs by construction. Numerical validation and comparison to macrogeometry methods are independent checks. This is a standard, non-circular analytic derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the established exact solution for the magnetic field of a single uniformly magnetized rectangular prism, with no new free parameters, axioms beyond standard magnetostatics, or invented entities introduced.

axioms (1)

domain assumption The magnetic field produced by a single uniformly magnetized rectangular prism is known exactly.
This is the explicit basis for extending to the periodic array as stated in the abstract.

pith-pipeline@v0.9.0 · 5408 in / 1135 out tokens · 34214 ms · 2026-05-10T14:49:17.884149+00:00 · methodology

discussion (0)

Reference graph

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