On the Role of Demagnetizing Tensors in Arbitrary Orientations of General Ellipsoid: Implications for MRI Safety Assessment

Tomppa Pakarinen

REVIEW 3 major objections 6 minor 23 references

On the Role of Demagnetizing Tensors in Arbitrary Orientations of General Ellipsoid: Implications for MRI Safety Assessment

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Rotating the correct demagnetizing tensor gives exact force and torque for arbitrarily oriented ellipsoids, and ties saturation behavior to the MRI magic angle.

desk verdict The tensor-rotation core and SVD area-projection are solid, but Eq. (12) misstates the translational force for arbitrary orientations, so the paper's main MRI-safety claim needs reworking. read the letter →

arxiv 2505.21761 v3 pith:P47KZQFD submitted 2025-05-27 physics.med-ph

Tomppa Pakarinen This is my paper

classification physics.med-ph

keywords demagnetizingtensorfactorsellipsoidMRIsafetytranslationalforcemagnetictorquemagicanglesingularvaluedecomposition

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

This paper argues that for ellipsoidal objects, the demagnetizing tensor, which encodes how shape and orientation alter the magnetization response, can be rotated directly into any new orientation when the original solution comes from the Poisson equation. That one move turns the usual single-axis trigonometric expressions for translational force and torque on spheroids into full tensor formulas valid for arbitrary three-dimensional rotations and for general ellipsoids with three unequal axes. The paper also tests a practical area-projection approximation, implemented with singular value decomposition, and finds it tracks the exact tensor rotation across prolate and oblate spheroids with fair accuracy. The broader payoff is a simpler route to MRI safety estimates for implants and foreign objects, and a new appearance of the well-known 54.7° magic angle as the orientation where prolate spheroid saturation-field curves converge.

What carries the argument

The load-bearing object is the demagnetizing tensor $\mathcal{N}$, a rank-2 tensor relating the internal demagnetizing field to magnetization; its principal-axis values are the demagnetizing factors $N_{ii}$, which sum to one. The paper's central mechanism is the transformation $\mathcal{N}' = R\mathcal{N}R^T$, valid for exact Poisson-derived solutions, which lets orientation dependence be handled by rotating the tensor rather than re-solving the boundary-value problem. The paper pairs this with the orthogonal-area projection approximation, in which the demagnetizing factor along the field is taken as the ratio of the perpendicular projected area to the total projected area; for ellipsoids the projection areas are computed from the singular values of the rotated shape matrix. These two pieces carry the derivation of the generalized force and torque formulas and the saturation-field calculation.

What would settle it

Compute the translational force on a general ellipsoid, such as one with axes 2, 5, and 1, in a uniform magnetic field with a known field gradient whose direction is not parallel to the field, at an orientation where the magnetization is oblique, and check whether the measured force has a component perpendicular to the gradient; equation (12) would fail if it does.

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Extended reading notes

Core claim

The central claim, stated on the paper's own terms, is that direct tensor rotation $\mathcal{N}' = R\mathcal{N}R^T$ with $R \in SO(3)$ is exact for general ellipsoids whose demagnetizing tensor is known in principal axes, so the vector forms $\mathbf{F}'_{\mathrm{trans}} = (V/\mu_0)\{(I+\chi\mathcal{N}')^{-1}\chi\}\mathbf{H}_0|\nabla\mathbf{B}|$ and $\mathbf{T}' = \mu_0 V\{(I+\chi\mathcal{N}')^{-1}\chi\mathbf{H}_0\}\times\mathbf{H}_0$ reproduce the standard single-axis results when specialized to spheroids and extend them to arbitrary three-dimensional rotations. The paper verifies the rotation numerically against direct numerical integration of the surface integral for the scalar potential, and uses the same reference to show that the area-projection approximation remains fairly accurate for spheroids. It also reports that for prolate spheroids, tensor-rotation saturation-field magnitudes converge across aspect ratios at the MRI magic angle of 54.7356°, which matches the root of the second-order Legendre polynomial that defines that angle.

Load-bearing premise

The paper's generalized force formula treats the force as a single number along the field-gradient direction, which is exact only when the induced magnetization points along that direction; that alignment is not guaranteed in arbitrary three-dimensional orientations.

Editorial extensions

If this is right

Force and torque on a spheroid under single-axis rotation reduce exactly to the standard trigonometric expressions, confirming the tensor form as a proper generalization.
For a general ellipsoid, one no longer needs separate formulas for each rotation plane; a single tensor rotation covers every orientation.
The area-projection method, extended by singular value decomposition, gives fair estimates with $R^2$ roughly between 0.86 and 0.97 for prolate and oblate spheroids at arbitrary orientations, which is enough for screening-level MRI safety estimates.
Saturation-field magnitudes computed by tensor rotation all pass through the same value at 54.7356°, coupling the familiar magic angle to shape-anisotropy independence at that orientation.

Reading between the lines

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

The exactness of $\mathcal{N}' = R\mathcal{N}R^T$ for uniformly magnetized ellipsoids suggests the same rotation recipe could apply to any body with a well-defined average demagnetizing tensor; verifying this numerically for non-ellipsoidal shapes would be a natural follow-up.
The magic-angle convergence makes a concrete prediction: at 54.7°, prolate spheroids of different aspect ratios should show nearly identical saturation-field behavior, a fact that an experiment with calibrated ellipsoids could confirm directly.
Equation (12) is only tested in the field-aligned case; testing it with a field gradient oblique to the applied field would clarify whether the scalar projection is a genuine limitation or an acceptable screening approximation.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper is worth a look for two genuinely useful pieces: the demonstration that rotating an ellipsoid's demagnetizing tensor via R N R^T is exact (validated against Osborn's formulas), and the SVD-based area-projection method that extends Bahl's trick to arbitrary 3D rotations. Both are clearly explained and the numerical checks are honest. The tensor rotation property itself is not new—Della Torre and Bort-Soldevila cover it—but the paper applies it cleanly in the MRI safety setting.

The magic angle observation is a nice curiosity: the saturation field curves for prolate spheroids converge at 54.7°, and the trace condition makes that unsurprising, but it is a neat connection.

Now the problem. Equation (12) claims to generalize the translational force to arbitrary orientations, but it is wrong as written. The true magnetostatic force on a uniformly magnetized body with a unidirectional gradient is F = V (M·ĥ)|∇B| ĥ, the projection of M onto the gradient direction. Equation (12) instead computes (V/μ0) M |∇B|, a vector along M. For a prolate spheroid at 45°, M and H0 differ by enough that this overestimates the force magnitude by roughly a third. The units also do not work with H0—this only makes sense if H0 is intended to be B0, and even then the direction error persists. The paper's comparison in Fig. 5 is likely done at orientations where M happens to align with the field, so the "exact match" claim does not test the actual claim. The torque formula (13) is fine, because it uses m×B directly.

So the tensor rotation result stands, the area-projection approximation is reasonable for spheroids, and the torque generalization is valid. The translational-force generalization, one of the four stated contributions, is not. That is a load-bearing flaw for the MRI safety framing.

The paper is honest about its scope and cites the relevant prior work, including the exact tensor rotation. No fitting or circularity in the validation. The SVD derivation is abbreviated—Eq. (20) is asserted more than derived—but the convex hull check supports it.

Who gets value: researchers working on quick demagnetizing-factor estimates, and anyone troubled by the standard scalar force formula in MRI safety. It deserves a serious referee because the core is correct and the error is fixable, but the referee should push for a corrected force formula and a public implementation before publication.

My recommendation: send it to review, but make clear the translational-force claim needs to be reworked or dropped.

Referee Report

3 major / 6 minor

Summary. This manuscript studies the behavior of demagnetizing tensors of general ellipsoids under arbitrary three-dimensional rotations, with applications to MRI safety assessment. The author shows that an exact Poisson-derived demagnetizing tensor transforms as N' = R N R^T with R in SO(3), validates this numerically against Osborn's formulas, and then uses the rotated tensor to propose generalized tensor expressions for translational force and torque. The paper also introduces an SVD-based area-projection approximation for demagnetizing factors, validates it against convex-hull projections and rotated exact tensors, and reports an empirical convergence of saturation-field magnitudes at the MRI magic angle for prolate spheroids.

Significance. The direct tensor rotation result is solid, parameter-free, and useful: it follows from the tensor nature of N and is numerically validated against independent exact solutions. The SVD area-projection approximation is clearly framed as approximate and is checked against two independent references (convex hull areas and rotated Osborn tensors) with reported R^2 values in the 0.86-0.97 range. The torque generalization is consistent with standard magnetostatics. However, the translational-force generalization in Eq. (12) is not a correct vector expression, and the magic-angle convergence is stated without a derivation. If these issues are corrected, the paper would be a useful practical contribution to MRI safety approximations.

major comments (3)

[Methods C, Eq. (12)] The translational-force formula in Eq. (12) is not a valid vector generalization and is the main load-bearing issue. For a uniformly magnetized body in a magnetostatic field, F = V (M·∇)B; with a unidirectional gradient ∇B = g ĥ this reduces to F = V (M·ĥ) g ĥ, i.e., the force is along the gradient direction, not along M. Equation (12) instead gives a vector proportional to (I + χN')^{-1}χ H0, which is M. These differ for arbitrary orientations: for a prolate spheroid of aspect ratio 10 at θ = 45°, the in-plane rotated tensor gives M_z/|M| ≈ 0.74, so using the norm of Eq. (12) overestimates the true force magnitude by roughly 35%. Figure 5 compares Eq. (12) with the scalar single-axis formula (8), which is the z-component of the force; if the plot uses the norm of Eq. (12), the agreement is not testing the formula as written, and if it uses the z-component, Eq. (12) should be written as the projection onto the gradient axis. The units of Eq. (12) are also inconsistent as printed (H0 in A/m and |∇B| in T/m with the V/μ0 prefactor yield A²), indicating that the intended expression along the gradient direction is V/μ0 (I+χN')^{-1}χ B0 |∇B|; the vector-direction issue persists regardless of that typo. The torque formula (13) does not suffer from this problem because it uses m×B directly.
[D.1, Eq. (20)] The derivation of the SVD area formula is too abbreviated to be independently reproduced. In particular, the statement that '√σ_i are the eigenvalues of collapsed (P_nk M'^{-1})^{-1}' is not self-consistent: singular values of a 2×2 projected matrix are not generally the eigenvalues of that inverse expression, and the 'dimension erasing' operation is not defined. The authors should provide the explicit 2×2 matrix whose SVD is computed, the exact relation between its singular values and the semiaxes of the projected ellipse, and a step-by-step derivation of A_P = π/√(σ11 σ22). The numerical validation against convex hulls suggests the formula is correct, but the reproducibility claim requires a precise statement.
[D, Saturation Field Magnitude, Fig. 6] The claimed convergence of the tensor-rotation saturation field to the magic angle is not supported by a derivation. The text states the result and shows the convergence in Fig. 6, but no equation for H_sat as a function of (N', χ) is given, and it is not shown analytically why all prolate aspect ratios cross at 0.955317 rad. Since this is listed as the paper's fourth main contribution, the authors should either derive the saturation-field expression from Eq. (13) and prove the common intersection point, or present the numerical computation with sufficient detail (including the saturation criterion) for the claim to be checked.

minor comments (6)

[Abstract, Introduction] The word 'exserted' appears several times; it should be 'exerted'.
[Results B] The sentence comparing the generalized solutions to 'expression in (7) and (9)' should refer to Eqs. (8) and (10), since those are the single-axis trigonometric forms being compared.
[Eqs. (7), (8), (12), (13)] The notation mixes scalars and vectors: B0 is a scalar in Eqs. (7) and (8), while H0 is a vector in Eqs. (12) and (13); please define the orientation of B0/H0 and the gradient direction for all force expressions.
[Fig. 5 caption] Specify whether the plotted quantity for the tensor method is the force component along the gradient or the norm; this is essential for interpreting the comparison.
[D.1] The sentence 'only σ22 is of interest, since 1/√σ11 = |u_minor| under all rotations' is unclear; define u_minor and justify the claim.
[Results A, Fig. 4] The discussion of the disparity when a/b→1 is ambiguous because the sphere limit has N=1/3 for every orientation; state explicitly how that limit is handled in the R² computation.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation chain is validated against independent exact solutions and external analytic formulas, with no fitted parameters or load-bearing self-citations.

full rationale

The paper's central claims are (1) direct rotation of the exact demagnetizing tensor, N' = RNR^T, is valid for general ellipsoids; (2) an area-projection/SVD approximation can reproduce demagnetizing factors under rotations; (3) tensor forms for translational force and torque generalize the single-axis trigonometric expressions; and (4) saturation field magnitudes for prolate spheroids converge at the magic angle. None of these claims is obtained by fitting a parameter to the quantity it later 'predicts'. The tensor rotation is checked against numerical surface integration of Poisson's equation (Fig. 1) and against Osborn's tabulated basis solutions [17]; the SVD projection method is compared to the exact rotated-tensor result (Figs. 3-4), not to data used to build it. The force and torque expressions are derived algebraically from M = chi (I + chi N')^{-1} H0 and compared to the standard trigonometric formulas of Schenck [13] and Abbot et al. [14]; the agreement for spheroids is a mathematical reduction, not a post hoc fit. The magic-angle convergence at 54.7356 degrees emerges from the tensor saturation-field expressions and is independent of the Legendre-polynomial derivation. The references are to external works (Osborn, Bahl, Schenck, Abbot, Della Torre); there are no load-bearing self-citations. Even if Eq. (12) has a physically questionable vector-direction assumption or inconsistent units, that is a correctness or validity concern, not circularity: the claim is not made true by definition or by a self-citation chain.

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

The paper introduces no fitted parameters or new physical entities. Its results follow from classical magnetostatics of uniformly magnetized ellipsoids, using Osborn's exact solutions as an external benchmark.

assumptions (3)

domain assumption Uniform magnetization within the ellipsoid (∇·M = 0 inside), so the volume term in Eq. (3) vanishes.
Standard result for ellipsoids; stated in Methods after Eq. (1).
standard math Demagnetizing tensor transforms as a rank-2 tensor under SO(3) rotation, 𝒩' = R𝒩R^T.
Invoked in Eq. (11) and validated numerically in Fig. 1 for a general ellipsoid.
domain assumption The trace of the demagnetizing tensor equals 1, Σ Nii = 1.
Used implicitly to show that saturation field magnitudes for all prolate spheroids converge at the magic angle; see Eq. (5) and the discussion section.

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Cite this review

Pith. "Pith review of On the Role of Demagnetizing Tensors in Arbitrary Orientations of General Ellipsoid: Implications for MRI Safety Assessment." pith.science (2026). https://pith.science/paper/P47KZQFD

@misc{pith2026250521761,
  author       = {Pith},
  title        = {Pith review of: On the Role of Demagnetizing Tensors in Arbitrary Orientations of General Ellipsoid: Implications for MRI Safety Assessment},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P47KZQFD}},
  note         = {Machine review of arXiv:2505.21761}
}

read the original abstract

This work explores the behaviour of demagnetizing tensors for general ellipsoids under arbitrary rotations in homogeneous magnetic fields. The work is motivated by the concerns in magnetic resonance imaging safety and their practical evaluation in clinical environments. Whereas demagnetizing tensor is a well-defined concept in the principal axes, its transformation under three-dimensional reorientation is often overlooked - a justifiable omission for solutions derived from Poisson equation, where the tensor can be directly rotated. However, this does not hold for common approximations, where such tensor is not explicitly defined. This work demonstrates the validity of directly rotating the orthogonal basis solutions, derived from Poissons equation, and uses the procedure to evaluate a practical approximation, based on orthogonal area-projections. The tensor rotation approach is also applied to generalize force and torque calculations for ellipsoids under three-dimensional re-orientation. The results show an exact match for translation force and torque when compared to the standard single axis rotation. Additionally, a unique connection to the well-known MRI magic angle is found as the point of convergence for prolate spheroid aspect ratios while solving the corresponding saturation field magnitudes. Finally, the evaluated approximate method was demonstrated to perform fairly across prolate and oblate spheroids. Similar approximations might extend to irregular shapes, but numerical validation would likely remain preferable due to the complexity of internal field distributions.

Figures

Figures reproduced from arXiv: 2505.21761 by the authors.

**Figure 5.** Translational force and torque comparison to expression in (7) and (9). Corresponding physical constants and field strengths were matched with [14]. The force was computed here with setting the field gradient strength to unity. The results show the generalized solutions for torque and translational force approach the trigonometric single axis method for 𝜒 ≫ 1, as expected. D. Saturation Field Magnitude Since torque … view at source ↗

**Figure 6.** Saturation field strength for prolate spheroid, computed using the tensor rotation method against the expression in Abbot et al. The results differ in non-orthogonal orientations by increasing magnitude with higher aspect ratios. Note that the tensor method solutions convergence to a single point at 0.955317 𝑟𝑎𝑑 = 54.7356º. The most notable caveat in [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗

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Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Griffiths and R

D. Griffiths and R. College, Introduction to Electrodynamics, New Jersey: Prentice Hall, 1999

work page 1999

[2] [2]

Simple and approximate expressions of demagnetizing factors of uniformly magnetized rectangular rod and cylinder,

M. Sato and Y. Ishii, “Simple and approximate expressions of demagnetizing factors of uniformly magnetized rectangular rod and cylinder,” Journal of Applied Physics, vol. 66, no. 2, pp. 983 -985, 1989

work page 1989

[3] [3]

An analytical computation of magnetic field generated from a cylinder ferromagnet,

T. Taniguchi, “An analytical computation of magnetic field generated from a cylinder ferromagnet,” Journal of the Physical Society of Japan, vol. 87, no. 4, p. 044802, 2018

work page 2018

[4] [4]

Fourier space derivation of the demagnetization tensor for uniformly magnetized objects of cylindrical symmetry,

F. Lang and S. Blundell, “Fourier space derivation of the demagnetization tensor for uniformly magnetized objects of cylindrical symmetry,” Journal of Magnetism and Magnetic Materials, vol. 401, pp. 1060-1067, 2016

work page 2016

[5] [5]

Demagnetizing Factors for Rectangular Prisms,

D. Chen, E. Pardo and A. Sanchez, “Demagnetizing Factors for Rectangular Prisms,” IEEE Transactions on Magnetics, pp. 2077 - 2088, 2005

work page 2005

[6] [6]

Magnetic resonance imaging: fundamental safety issues.,

G. Durbridge, “Magnetic resonance imaging: fundamental safety issues.,” Journal of orthopaedic & sports physical therapy, vol. 41, no. 11, pp. 820-828, 2011

work page 2011

[7] [7]

Standard Test Method for Measurement of Magnetically Induced Displacement Force on Medical Devices in the Magnetic Resonance Environment.,

i. ASTM, “Standard Test Method for Measurement of Magnetically Induced Displacement Force on Medical Devices in the Magnetic Resonance Environment.,” ASTM International, West Conshohocken, PA, 2021. 7

work page 2021

[8] [8]

ICNIRP Statement —Guidelines on Limits of Exposure to Static Magnetic Fields,

I. C. o. N. -I. R. P. ICNIRP, “ICNIRP Statement —Guidelines on Limits of Exposure to Static Magnetic Fields,” Health Physics, vol. 96, no. 4, pp. 504-514, 2009

work page 2009