Chemistry Ferromagnetism Fe Higher Moment Than Ni

The iron-nickel moment comparison

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 tiny magnetic fields produced by each atom's electrons. One of the standard facts about ferromagnetic metals is that their saturation magnetizations differ: the maximum magnetic moment per atom that iron can achieve is larger than that of nickel. In the Recognition Science framework, this comparison is not merely asserted; it is recorded as a proved theorem in a machine-checked library of formal theorems.

The theorem fe_higher_moment_than_ni states that the saturation moment of iron (atomic number 26) is greater than that of nickel (atomic number 28). The library defines these moments explicitly: iron's saturation moment is set at 2.22 Bohr magnetons per atom, while nickel's is set at 0.61. The proof is a direct computation from these definitions, verified by the library's kernel. This means the claim is not an empirical measurement or a hand-waving estimate; it is a formal consequence of the library's chosen values.

In Recognition Science, the framework models ferromagnetism through a chain of mechanisms: the exchange interaction arising from the Pauli exclusion principle, an eight-tick coherence structure in d-orbital degeneracy, the Stoner criterion for when ferromagnetism occurs, and the Curie temperature above which thermal fluctuations destroy magnetic order. The library also proves related facts, such as that cobalt has the highest Curie temperature among the three metals, and that gadolinium has the highest saturation moment of all the elements it lists. These theorems build a consistent, machine-checked picture of the basic phenomenology of ferromagnetism.

What the theorem does not claim is important. It does not derive the saturation moment values from first principles; the values 2.22 and 0.61 are chosen as definitions in the library, not proved from deeper physics. The theorem only establishes the ordering between the two numbers as defined. It also does not claim that iron's moment is the highest among all ferromagnets; the library separately proves that gadolinium's moment exceeds iron's. Finally, it makes no statement about the physical mechanism that produces the difference in moments, only about the numerical relationship between the two defined quantities.

/-- Fe has higher moment than Ni. -/
theorem fe_higher_moment_than_ni :
    saturationMoment 26 > saturationMoment 28 := by
  simp only [saturationMoment]
  norm_num

/-- Saturation magnetic moment per atom (Bohr magnetons). -/
def saturationMoment : ℕ → ℝ
| 26 => 2.22  -- Fe
| 27 => 1.72  -- Co
| 28 => 0.61  -- Ni
| 64 => 7.63  -- Gd
| _ => 0

What this page does not claim

The saturation moment values are not derived from first principles; they are chosen as definitions. Iron's moment is not claimed to be the highest among all ferromagnets; gadolinium's is higher. The theorem says nothing about the physical mechanism behind the moment difference.

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)

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