Test 13-coercivity-grain — Coercivity Peaks at a Grain Size
Supplied working report; original verdicts not validated.
Review status: Needs review. This is an archival source note, not a passed proof.
report.md
# Test 13-coercivity-grain — Coercivity Peaks at a Grain Size Test of the hypothesis in *Original Theories Vol XIII*, `theories/coercivity-grain-single-domain.html` (Theory 13 of 62, "Magnetic Materials"). ## Source **Claim (verbatim):** "Coercivity is maximised when the grain size approaches the single-domain size, falling for both smaller and larger grains." **Mechanism stated in the source (verbatim):** "Below single-domain size, thermal fluctuations erase magnetisation; above it, domain walls move easily and demagnetise the grain at low field." **Predictions (verbatim):** - P1 — "Coercivity should peak at the single-domain size." - P2 — "Both finer and coarser grains should be softer." **Refutation (verbatim):** "Monotonic coercivity with grain size would remove the optimum." Operational form used for the test (per test brief): above the single-domain size D_sd, H_c ∝ 1/D with the wall-pinning ceiling 2K/(μ₀M_s); below D_sd, H_c ∝ D⁶ (Herzer's random-anisotropy law). So H_c(D) should have a maximum at D ≈ D_sd. ## Why probable - **Coarse side (D > D_sd).** Grains contain domain walls; coercivity is set by wall pinning. Pinning theory gives H_c = p_w·√(A·K)/(μ₀·M_s·D) — i.e. H_c ∝ 1/D — so coarser grains are softer, and H_c rises as D approaches D_sd. The ideal ceiling at D = D_sd is the anisotropy field H_A = 2K/(μ₀M_s). Electrical-steel data show H_c increasing as grain size decreases (Landgraf et al. 2011); sintered Nd-Fe-B follows H_c ∝ 1/ln(D²), rising as D shrinks (Li et al. 2009). - **Fine side (D < D_sd).** In exchange-coupled nanocrystalline soft magnets the local magnetocrystalline anisotropy is averaged out over the exchange length L_ex = √(A/K): H_c = p_c·K⁴·D⁶/(J_s·A³) (Herzer 1990, p_c ≈ 0.13). This D⁶ drop means finer grains are softer — the "softer below the peak" half of the claim. - **The crossover** between the rising 1/D branch and the falling D⁶ branch lies near D_sd = 72√(A·K)/(μ₀M_s²) (Kittel 1949 estimate of the critical single-domain diameter; as quoted e.g. in Lyubina 2006, eq. 1.11). Hence a maximum is expected near D_sd — the hypothesis is a plausible synthesis of two well-established regimes. ## Method ### Formulas 1. Single-domain size (spherical-particle estimate, Kittel 1949; Lyubina 2006 eq. 1.11): **D_sd = 72·√(A·K₁)/(μ₀·M_s²)**, with A exchange stiffness (J/m), K₁ first anisotropy constant (J/m³), M_s saturation magnetisation (A/m). 2. Coarse branch (D > D_sd): **H_c(D) = [2K₁/(μ₀M_s)]·(D_sd/D)** — the 2K/(μ₀M_s) wall-pinning ceiling, declining as 1/D. 3. Fine branch (D < D_sd): **H_c(D) = p_c·K₁⁴·D⁶/(J_s·A³)** (Herzer random-anisotropy law, p_c = 0.13; J_s = μ₀M_s), valid for D < L_ex = √(A/K). 4. Composite: H_c(D) = min(branch 2, branch 3); its maximum is compared with D_sd. ### Material constants used (each with citation) | Material | μ₀M_s (T) | \|K₁\| (J/m³) | A (J/m) | Source | |---|---|---|---|---| | Fe (bcc) | 2.185 | 4.80×10⁴ | 2.07×10⁻¹¹ | Southampton micromagnetic thesis, Table C.1 | | Co (hcp) | 1.76 | 5.20×10⁵ | 3.0×10⁻¹¹ | nmag `two_cubes` example (OOMMF values); cf. Southampton thesis Table 2.3 | | Ni | 0.62 | 4.5×10³ | 7.2×10⁻¹² | Southampton thesis, Table C.1 (Table 2.3 lists 9×10⁻¹²) | | Fe₃O₄ | 0.61 | 1.36×10⁴ | 1.32×10⁻¹¹ | Southampton thesis Table C.1 (after Afremov & Panov 1998); A = 1.2×10⁻¹¹ "usually quoted" (Dantas & Gama 2010) | | Nd₂Fe₁₄B | 1.61 | 4.30×10⁶ | 7.7×10⁻¹² | Fischbacher et al. 2017, Table 1 (after Hock 1988; Durst & Kronmüller 1986) | | Fe–Si (Herzer canonical) | 1.20 | 8×10³ | 1.0×10⁻¹¹ | Herzer, IEEE Trans. Magn. 26 (1990) 1397; as quoted in Tsepelev et al. 2026 | | Fe–6.5 wt% Si | 1.85 | 1.8×10⁴* | 1.7×10⁻¹¹* | B_s = 1.85 T from Inoue et al. 2014 Table 3; *K₁, A typical micromagnetic values (range 8–20 kJ/m³, 1–2×10⁻¹¹ J/m) | | Permalloy Ni₈₀Fe₂₀ | 1.00 | ≈0 | 1.3×10⁻¹¹ | nmag example; Southampton thesis; A = 1.05×10⁻¹¹ (Smith et al., via Dantas & Gama) | ### Published measured series extracted (data/published_series.csv) - **Fe₃O₄ cube-like nanoparticles** (Li et al., Sci. Rep. 7:9894, 2017): H_c(D) for dp = 9.6–287 nm; H_c ≈ 0 at 9.6 nm (superparamagnetic), maximum 190 Oe at the single-domain/multi-domain critical size **75 ± 3 nm**, decreasing for 130–287 nm; the authors state the initial H_c increase follows the **sixth power** of the size. - **Soft nanocrystalline alloys** (Inoue et al., J. Alloys Compd. 615 (2014) S2–S8, Table 3): (D, H_c) = FINEMET Fe₇₃.₅Si₁₃.₅B₉Nb₃Cu₁ (20 nm, 0.5 A/m); NANOPERM Fe₉₀Zr₇B₃ (13 nm, 5.8 A/m); Fe₈₅Si₂B₈P₄Cu₁ (16 nm, 5.8); Fe₈₆Si₁B₈P₄Cu₁ (19 nm, 2.8); Fe₈₂.₆₅Si₂B₉P₅Cu₁.₃₅ (15 nm, 1.1); Fe₈₃.₃Si₄B₈P₄Cu₀.₇ (10 nm, 7); Fe₈₄.₃Si₄B₈P₄Cu₀.₇ (17 nm, 10); Fe₈₃B₁₀C₄Si₂Cu₁ (15 nm, 4); amorphous Fe₇₈Si₉B₁₃ (2.6 A/m); Fe–6.5%Si steel 45 A/m (D not tabulated). Plus FINEMET-type Fe₇₂.₅Cu₁Nb₂Mo₁.₅Si₁₄B₉ (Tsepelev et al., Nanomaterials 16:922, 2026): D = 7–9 nm, H_c = 0.41 A/m. The classic Herzer compilation (Handbook of Magnetic Materials vol. 10, p. 415 (1997); figure reproduced in Schaefer's ESM-2007 slides) shows the same family of curves rising as D⁶ through the nanocrystalline range, peaking in the fine- crystalline range (~30–100 nm) and falling as 1/D for coarse grains. - **Nd–Fe–B**: sintered magnets (Li, Ohkubo, Hono & Sagawa, JMMM 321 (2009) 1100): peak μ₀H_c = 17 kOe at D ≈ 4.5 μm with H_c ∝ 1/ln(D²) above it; melt-spun magnets have ideal grain sizes 20–100 nm and the single-domain size is quoted as 0.3 μm (Branagan, PhD thesis, Iowa State 1995, OSTI 42819). ## Results ### Computed single-domain sizes | Material | D_sd (nm) | Notes | |---|---|---| | Fe | 18.9 | literature ~15–25 nm ✓ | | Co | 115 | literature ~40–100 nm (shape-sensitive) | | Ni | 42.4 | literature ~40–55 nm ✓ | | Fe₃O₄ | 103 (alt. constants: 91) | measured transition 75 ± 3 nm (Li 2017) | | Nd₂Fe₁₄B | 201 (0.20 μm) | literature 0.3 μm (Branagan 1995) | | Fe–Si (Herzer canonical) | 17.8 | Fe–6.5%Si variant: 14.6 | | Permalloy | undefined | K₁ ≈ 0 → formula degenerates; single-domain size is then set by shape anisotropy | ### Analytical branches For Fe–Si (K₁ = 8 kJ/m³, A = 10⁻¹¹ J/m, J_s = 1.2 T): the D⁶ branch gives 14.0 A/m at D_sd = 17.8 nm (and 0.44 A/m at 10 nm — matching measured FINEMET-type H_c of 0.4–0.5 A/m at 7–10 nm). The 2K/(μ₀M_s) ceiling at D_sd is 1.33×10⁴ A/m. The composite curve (min of the two branches) **peaks at D = 47.4 nm = 2.66 × D_sd**, i.e. the two hypothesised branches do produce a maximum near D_sd (within the factor-3 tolerance). ### Measured vs predicted (factor-3 test) | Series | Measured peak | Computed D_sd | Ratio | Position test | Coarse branch declines? | |---|---|---|---|---|---| | Fe₃O₄ nanoparticles | 75 ± 3 nm (Li 2017) | 91–103 nm | 1.2–1.4 | **PASS (×3)** | yes (130–287 nm softer) | | Fe–Si-based nanocrystalline | rising branch 7–20 nm, 0.4–10 A/m; peak in fine-crystalline range ~30–100 nm (Herzer figure) | 17.8 nm (composite peak 47 nm) | ~2–3 | **PASS (×3), peak region qualitatively** | yes (coarse FeSi steels: H_c rises as D falls, Landgraf 2011; Fe–6.5%Si 45 A/m) | | Nd–Fe–B sintered | 4.5 μm (Li 2009) | 0.20 μm (lit. 0.30 μm) | **15–22** | **FAIL (>×3)** | yes above peak (H_c ∝ 1/ln D²); below 4.5 μm H_c drops from fcc-Nd oxide, not domain physics | ## Verdict **PARTIAL** A non-monotonic H_c(D) with a maximum and a declining coarse branch is confirmed in every published series examined, and for Fe₃O₄ the measured peak (75 ± 3 nm) matches the computed D_sd (91–103 nm) within a factor 1.2–1.4, with the D⁶ rise reproduced explicitly. For the Fe–Si nanocrystalline family the peak lies within a factor ~2–3 of D_sd. However, for Nd–Fe–B the measured peak sits 15–22× above D_sd, and below D_sd nanocrystalline Nd–Fe–B does **not** soften (the random-anisotropy D⁶ branch and the "thermal fluctuations erase magnetisation" mechanism do not apply to hard magnets), so the claim is not universally supported at the stated factor-3 tolerance. The peak exists but its position/scaling is off by more than the stated factor in one of the three main material families → PARTIAL. ## Limitations - Fe₃O₄ per-size H_c values (Supplementary Table S1 of Li et al. 2017) were not accessible; the peak value and position are taken from the published text, the declining branch only qualitatively. - The Herzer FINEMET peak position (~30–100 nm) is read qualitatively from the reproduced Handbook figure (Schaefer ESM-2007 slides); no tabulated peak points were found in open sources. - The composite-branch peak *magnitude* (≈5 kA/m for Fe–Si) overshoots the measured FINEMET peak because the 2K/(μ₀M_s) ceiling is the ideal anisotropy field; real wall pinning sits 1–2 orders below the ceiling (p_w ≪ 1). Position, not magnitude, is the claim's testable content. - The D⁶ law is valid only for D < L_ex = √(A/K) (≈35 nm for Fe–Si); it was extrapolated slightly beyond L_ex to locate the branch crossover. - K₁ and A for Fe–6.5%Si are typical literature values, not from a single cited measurement; A for Ni and Fe₃O₄ varies by 2–3× across sources, shifting D_sd by √(A·K) ≈ 1.5–1.7×. - Permalloy (K₁ ≈ 0) makes the D_sd formula degenerate; such materials were excluded from the peak-position test. - For hard magnets (Nd–Fe–B) the hypothesis's below-D_sd mechanism (softening) is contradicted by data; the measured sub-4.5 μm drop in sintered magnets is a processing (oxidation) artifact, and nanograined Nd–Fe–B retains ≈1 T coercivity. ## References 1. Hypothesis source (verbatim text): `[local provenance path] Theories Vol XIII - 2026-10-30/theories/coercivity-grain-single-domain.html` 2. Kittel 1949 single-domain estimate as quoted in: J. Lyubina, *Nanocrystalline Fe–Pt alloys*, PhD thesis, TU Dresden (2006), eq. (1.11) — https://www.osti.gov/etdeweb/servlets/purl/20897251 3. Micromagnetic material constants tables: Southampton thesis, Tables 2.3 & C.1 — https://www.southampton.ac.uk/~rpb/thesis.pdf 4. nmag `two_cubes` example (Co & permalloy, OOMMF values) — https://nmag.readthedocs.io/en/latest/_downloads/two_cubes.py 5. C. C. Dantas & A. M. Gama, *Micromagnetic simulations of spinel ferrite particles* (magnetite A = 1.2×10⁻¹¹ J/m; permalloy A = 1.05×10⁻¹¹, Smith et al.) — https://ar5iv.labs.arxiv.org/html/1005.3169 6. J. Fischbacher et al., *On the limits of coercivity in permanent magnets* (Nd₂Fe₁₄B constants, Table 1) — https://ar5iv.labs.arxiv.org/html/1708.01880 7. G. Herzer, *Grain size dependence of coercivity and permeability in nanocrystalline ferromagnets*, IEEE Trans. Magn. 26 (1990) 1397–1402 (D⁶ law, p_c ≈ 0.13); abstract record — https://scholar.google.hr/citations?view_op=view_citation&hl=zh-CN&user=M2x-nZUAAAAJ&citation_for_view=M2x-nZUAAAAJ:u5HHmVD_uO8C 8. G. Herzer, *Nanocrystalline soft magnetic materials*, Phys. Scr. T49A (1993) 307 (grain sizes 10–15 nm) — https://beta.iopscience.iop.org/article/10.1088/0031-8949/1993/T49A/054/meta 9. G. Herzer, *Nanocrystalline soft magnetic alloys*, in Handbook of Magnetic Materials, vol. 10, p. 415 (Elsevier, 1997) — figure (H_c vs D, 1/D and D⁶ branches) reproduced in: R. Schäfer, *Coercivity and grain size*, ESM-2007 Cluj lecture slides — https://magnetism.eu/esm/2007-cluj/slides/schaefer-slides.pdf 10. V. S. Tsepelev, K. Wu & N. P. Tsepeleva, Nanomaterials 16:922 (2026) (FINEMET-type D = 7–9 nm, H_c = 0.41 A/m) — https://www.mdpi.com/2079-4991/16/15/922 11. A. Inoue, F. L. Kong, Q. K. Man, B. L. Shen, R. W. Li & F. Al-Marzouki, *Development and applications of Fe- and Co-based bulk glassy alloys…*, J. Alloys Compd. 615 (2014) S2–S8, Table 3 (open mirror) — https://smse.seu.edu.cn/_upload/article/files/71/73/6e4da25a42d89da8c47cbef7ab1f/b63cef2e-f032-4a85-b4e0-b52728d74b49.pdf 12. Q. Li, C. W. Kartikowati, S. Horie, T. Ogi, T. Iwaki & K. Okuyama, *Correlation between particle size/domain structure and magnetic properties of highly crystalline Fe₃O₄ nanoparticles*, Sci. Rep. 7:9894 (2017), doi:10.1038/s41598-017-09897-5 — https://www.nature.com/articles/s41598-017-09897-5 13. W. F. Li, T. Ohkubo, K. Hono & M. Sagawa, *The origin of coercivity decrease in fine grained Nd-Fe-B sintered magnets*, J. Magn. Magn. Mater. 321 (2009) 1100–1105, doi:10.1016/j.jmmm.2008.10.032 — https://inis.iaea.org/records/yajdk-2d978 14. D. J. Branagan, *A metallurgical approach toward alloying in rare earth permanent magnet systems*, PhD thesis, Iowa State Univ. (1995), IS-T-1719, OSTI ID 42819 (Nd₂Fe₁₄B single-domain size 0.3 μm; sintered 10–20 μm; melt-spun 20–100 nm) — https://www.osti.gov/biblio/42819 15. F. J. G. Landgraf, J. R. F. da Silveira & D. L. Rodrigues, *Determining the effect of grain size and maximum induction upon coercive field of electrical steels*, J. Magn. Magn. Mater. 323 (2011) 2335–2339, doi:10.1016/j.jmmm.2011.03.034 — https://repositorio.usp.br/single.php?_id=002214548 16. B. Hofmann, T. Reininger & H. Kronmüller, *Influence of the Microstructure on the Magnetization Processes in Nanocrystalline Fe₇₃.₅Cu₁Nb₃Si₁₃.₅B₉*, phys. stat. sol. (a) 134 (1992) 247–261, doi:10.1002/pssa.2211340122 (H_c changes by orders of magnitude at fixed D ≈ 12 nm — a caveat to pure grain-size scaling) — http://archive.nstl.gov.cn/Archives/browse.do?action=viewDetail&articleID=4f74f19befa90a34 ## Files - `report.md` — this report - `code/coercivity_test.py` — computes D_sd, draws both analytical branches and the composite peak, overlays the published series, writes the figure - `data/published_series.csv` — extracted published data points with source notes - `figures/coercivity_vs_grainsize.png` — H_c vs grain size (log–log) - `scratch/` — downloaded source PDFs used for extraction (Schaefer ESM-2007 slides, Branagan thesis, Lyubina thesis, Southampton thesis, Inoue review, Li 2017 paper, Tsepelev 2026 paper, OSTI/ETDEWEB records) Working report — not peer reviewed. Educational information, not medical advice. AI-assisted test: DeepSeek.