Chemistry Ferromagnetism Fe Stoner Satisfied

Iron's Stoner condition

Ferromagnetism is the mechanism by which certain materials, such as iron, cobalt, and nickel, form permanent magnets and are attracted to magnets. It arises from the spontaneous alignment of atomic magnetic moments. The Stoner criterion is a simple condition that predicts when this alignment can occur: it requires the product of the exchange interaction strength (U) and the density of states at the Fermi level (D(E_F)) to be greater than 1. When U × D(E_F) > 1, the energy cost of aligning spins is overcome by the energy gain from exchange, and ferromagnetic order can set in.

In the Recognition Science framework's machine-checked library of formal theorems, the declaration fe_stoner_satisfied proves that for iron, this condition holds. The framework defines the Stoner interaction parameter for iron as 0.9 and the density of states at the Fermi level as 1.5. The proof calculates the product: 0.9 × 1.5 = 1.35, which is indeed greater than 1. This is a formal, verified result within the framework's ledger, a discrete record of recognition events. The declaration itself is a theorem, meaning it is a proved statement in the framework's system, not merely an assumption.

What this declaration does not claim is equally important. It does not prove that iron is ferromagnetic in the physical world. That is a separate statement, iron_ferromagnetic, which is also in the library. The Stoner criterion is a necessary condition for ferromagnetism, not a sufficient one. A material could satisfy the criterion yet fail to be ferromagnetic due to other factors. The declaration also does not derive the values 0.9 and 1.5 from first principles; these are chosen definitions within the framework, not results of a proof. It simply verifies that, given these defined values, the inequality holds.

The significance of this theorem is that it provides a formal, checkable instance of a physical criterion. It shows that the framework can encode and verify a standard result from condensed matter physics. The proof is a simple arithmetic calculation, but its value lies in being part of a larger, axiomatically clean system. It demonstrates that the framework's machinery can handle not just abstract mathematics but also concrete physical applications, even if the physical constants themselves are inputs rather than outputs.

/-- Fe satisfies Stoner criterion. -/
theorem fe_stoner_satisfied : stonerI_Fe * dos_Fe > 1 := by
  simp only [stonerI_Fe, dos_Fe]
  norm_num

/-- Stoner parameter for Fe (eV). -/
def stonerI_Fe : ℝ := 0.9

/-- Density of states for Fe at Fermi level (states/eV/atom). -/
def dos_Fe : ℝ := 1.5

What this page does not claim

It does not prove that iron is ferromagnetic in the physical world; that is a separate theorem. It does not derive the values 0.9 and 1.5 from first principles; they are defined constants. It does not claim the Stoner criterion is sufficient for ferromagnetism, only necessary.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Chemistry/Ferromagnetism.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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