Chemistry Ferromagnetism Ferromagnet Positive J

The positive exchange condition

Ferromagnetism is the mechanism by which certain materials, most famously 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. In a ferromagnet, these moments line up in the same direction over large regions, producing a net magnetic field. The alignment is not permanent in the sense of being unbreakable: above a material-specific temperature called the Curie temperature, thermal jostling overcomes the alignment and the material becomes merely paramagnetic, with moments pointing randomly.

The quantum-mechanical engine behind this alignment is the exchange interaction, which comes from the Pauli exclusion principle: two electrons cannot occupy the same quantum state, so electrons with parallel spins tend to avoid each other, reducing their mutual Coulomb repulsion. The strength of this interaction is quantified by an exchange constant, often denoted J. When J is positive, parallel alignment of neighboring spins is energetically favored, which is the precondition for ferromagnetism. When J is negative, the opposite alignment, antiferromagnetism, is favored. The sign of J is therefore the single most important switch in the physics of magnetic order.

In the Recognition Science framework, the machine-checked library of formal theorems contains a declaration named ferromagnet_positive_J. The framework models the exchange interaction through a function exchangeJ that assigns a numerical strength to each element by atomic number. The theorem states that for any element in the framework's list of ferromagnetic elements, which is exactly iron (Z=26), cobalt (Z=27), and nickel (Z=28), the exchange constant exchangeJ Z is greater than zero. In plain language, it establishes that the three canonical ferromagnets all have positive exchange constants, the necessary condition for their spins to align spontaneously.

The theorem is proved by direct computation from the definitions in the library, not by physical experiment. The library also records the specific values: exchangeJ is 10.0 for iron, 14.0 for cobalt, and 8.0 for nickel, all in arbitrary framework units. These values are definitional choices, not measurements. The theorem's content is the logical consequence of those choices, namely that all three are positive. It does not claim that these numbers match laboratory measurements, nor does it derive them from first principles. It is a consistency check on the model, not an empirical prediction.

The declaration also does not claim that positive J alone is sufficient for ferromagnetism. The framework's own docstring notes the Stoner criterion, U × D(E_F) > 1, where U is the exchange interaction strength and D(E_F) is the density of states at the Fermi level. A material could have positive J yet fail this criterion and remain non-magnetic. The theorem is a necessary condition, not a complete theory. What it gives the reader is a precise, machine-checked statement that the model's three canonical ferromagnets all satisfy the basic sign requirement, and nothing more.

/-- Ferromagnetic elements have positive exchange J. -/
theorem ferromagnet_positive_J (Z : ℕ) (h : Z ∈ ferromagneticElements) :
    exchangeJ Z > 0 := by
  simp only [ferromagneticElements] at h
  fin_cases h <;> simp only [exchangeJ] <;> norm_num

/-- Exchange energy J (meV). Positive J → ferromagnetic coupling. -/
def exchangeJ : ℕ → ℝ
| 26 => 10.0   -- Fe
| 27 => 14.0   -- Co
| 28 => 8.0    -- Ni
| _ => 0

/-- Stoner criterion: U × D(E_F) > 1 for ferromagnetism.
    U is exchange interaction, D(E_F) is density of states at Fermi level. -/
def stonerCriterion (U D_EF : ℝ) : Bool := U * D_EF > 1

What this page does not claim

The theorem does not claim that the exchange constant values are measured or derived from first principles. The theorem does not claim that positive J alone is sufficient for ferromagnetism, since the Stoner criterion must also be satisfied. The declaration does not claim that gadolinium or other rare earths are covered by this particular theorem.

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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