Anisotropy Is the Property That Makes a Magnet Permanent
Iron has a large moment and a high Curie temperature, and it is useless as a permanent magnet. What separates a magnet from a lump of ferromagnet is how firmly its magnetisation is anchored to a crystal direction.
Magnets and Motors · Part 2 — The Permanent Magnet · Chapter 4 · 18 min read
Iron has a large moment and a high Curie temperature, and it is useless as a permanent magnet. Its magnetisation turns whenever a field asks it to.
What separates a magnet from a lump of ferromagnet is not how strong it is. It is how firmly its magnetisation is anchored to a crystal direction.
0.048
Iron’s K₁, MJ/m³
4.9
Nd₂Fe₁₄B’s — a hundred times higher
15–30 %
Of theoretical coercivity actually achieved
0.2–0.3 µm
Single-domain grain size for NdFeB
4.1 — The property that makes a magnet permanent
- •Magnetocrystalline anisotropy (K₁, J/m³) — the energy difference between magnetising a crystal along its easy axis and along a hard axis. It arises from spin–orbit coupling: the electron spin is coupled to its orbital motion, the orbital shape is fixed to the lattice by the crystal field, and so the spin is indirectly bolted to the crystal.
- •Anisotropy field (μ₀H_A, tesla) — the field required to rotate the magnetisation coherently away from its easy axis, and the theoretical upper limit on coercivity.
μ₀H_A = 2K₁ / J_s
J_s is the saturation polarisation.
4.2 — Why the answer is always a rare earth
Spin–orbit coupling scales steeply with atomic number, and the 4f charge cloud is strongly aspherical — Nd³⁺ and Dy³⁺ are shaped like flattened or elongated ellipsoids rather than spheres. An aspherical charge cloud sitting in a crystal field has a strongly preferred orientation, and that preference is transmitted straight to the spin.
The 3d metals have the opposite problem: their orbital moment is largely quenched by the crystal field, because 3d orbitals stick out and interact strongly with their neighbours. Little orbital moment means little spin–orbit anchoring means little anisotropy.
Important
Iron’s K₁ is 0.048 MJ/m³. Nd₂Fe₁₄B’s is 4.9 — a hundred times higher. That ratio, not the moment, is what makes one a magnet and the other a transformer core.
4.3 — Brown’s paradox — nobody gets more than a third
Technical framing
Every real magnet achieves only 15 to 30 per cent of its theoretical anisotropy field. Magnetisation reversal does not happen by coherent rotation of a whole grain — it nucleates at a defect, a grain surface or a misaligned region, and then sweeps through.
Coercivity is therefore a microstructure property, not just a chemistry one, which is why two magnets of identical composition can differ by a factor of two.
4.4 — Domains, and why a magnet demagnetises itself
A uniformly magnetised block stores a great deal of energy in the external field it creates. It can lower that energy by splitting into domains pointing different ways, at the cost of the energy needed to build domain walls between them. The equilibrium is a balance between the two.
In plain English
Below a critical grain size the wall energy exceeds the saving, and splitting stops being worthwhile. The grain stays single-domain — and a single-domain grain can only reverse by rotating its whole magnetisation against the full anisotropy, which is hard. For Nd₂Fe₁₄B that critical size is about 0.2 to 0.3 µm.
Why this matters
This is why sintered magnet processing is what it is. The powder is milled to a few microns, aligned in a field, pressed and sintered — with the whole process aimed at small, well-isolated, well-aligned grains. Grain size and grain boundary chemistry are the coercivity levers, and both are metallurgy rather than composition.
4.5 — Grain boundary diffusion — using dysprosium only where it does work
Reversal nucleates at grain surfaces. So the high-anisotropy heavy rare earth is only needed at grain surfaces — not distributed through the bulk, which is what conventional alloying does.
4.5.1 — The process
Coat a sintered magnet in a Dy or Tb compound and heat-treat it, letting the heavy element migrate along grain boundaries and form a thin high-anisotropy shell on each grain.
The result: coercivity gains equivalent to 2 to 6 wt % bulk dysprosium, using well under 1 wt %, with almost no loss of remanence because the grain interiors stay pure Nd₂Fe₁₄B.
Important
It is the single most consequential process innovation in magnets in twenty years, and it exists because of a microstructural insight rather than a new material. Chapter 10 covers what it did to heavy rare earth demand.
Quick check: test yourself
1.Iron has a bigger moment than neodymium and a higher Curie temperature. Why is it not the magnet?
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2.Two magnets have identical composition and differ in coercivity by a factor of two. How?
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3.Why does grain boundary diffusion save so much dysprosium?
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Chapter summary
- ✓Anisotropy, not moment or Curie temperature, is what makes a magnet permanent — it is what anchors the magnetisation to a crystal direction.
- ✓It comes from spin–orbit coupling, which scales with atomic number and needs an aspherical charge cloud — hence a rare earth, since 3d orbital moments are quenched.
- ✓Brown’s paradox: every real magnet reaches only 15 to 30 per cent of its theoretical anisotropy field, because reversal nucleates at defects rather than rotating coherently.
- ✓Below about 0.2 to 0.3 µm a grain stays single-domain and can only reverse against the full anisotropy, which is why sintered processing chases small, aligned, isolated grains.
- ✓Grain boundary diffusion puts the heavy rare earth only where reversal starts — the most consequential magnet process innovation in twenty years, and a microstructural insight rather than a new material.
Frequently asked questions
Why is iron useless as a permanent magnet despite being strongly ferromagnetic?+
Because its magnetisation turns whenever a field asks it to. Being ferromagnetic means having a large moment and a high Curie temperature; being a permanent magnet means having the magnetisation anchored to a crystal direction so it resists turning. Iron’s magnetocrystalline anisotropy constant K₁ is 0.048 MJ/m³ against Nd₂Fe₁₄B’s 4.9 — a hundred times higher. Anisotropy, not strength, is what makes a magnet permanent.
Why does high anisotropy always come from a rare earth?+
Two reasons that compound. Spin–orbit coupling scales steeply with atomic number, and the 4f charge cloud is strongly aspherical — Nd³⁺ and Dy³⁺ are shaped like flattened or elongated ellipsoids rather than spheres. An aspherical charge cloud in a crystal field has a strongly preferred orientation, and that preference is transmitted straight to the spin. The 3d metals have the opposite problem: their orbital moment is largely quenched by the crystal field because 3d orbitals stick out and interact strongly with neighbours.
What is Brown’s paradox?+
Every real magnet achieves only about 15 to 30 per cent of its theoretical anisotropy field. The reason is that magnetisation reversal does not happen by coherent rotation of a whole grain — it nucleates at a defect, a grain surface or a misaligned region, and then sweeps through. Coercivity is therefore a microstructure property, not just a chemistry one, which is why two magnets of identical composition can differ by a factor of two.
How does grain boundary diffusion work, and why does it matter so much?+
Because reversal nucleates at grain surfaces, the high-anisotropy heavy rare earth is only needed at grain surfaces — not distributed through the bulk, which is what conventional alloying does. Grain boundary diffusion coats a sintered magnet in a Dy or Tb compound and heat-treats it so the heavy element migrates along grain boundaries and forms a thin high-anisotropy shell on each grain. The result is coercivity equivalent to 2 to 6 wt % bulk dysprosium using well under 1 wt %, with almost no loss of remanence because the grain interiors stay pure Nd₂Fe₁₄B.
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Magnets and Motors is an original educational series on permanent magnets and electric machines. Datasheet ranges, temperature coefficients and efficiency figures are representative standard-condition literature values, not measurements of a specific product, and sources differ on several. Always verify against the specific magnet and lamination datasheets in use before making design, procurement or certification decisions.