Materials Domain Wall Width2

Domain wall width

A domain wall is the transition layer in a magnetic material where the direction of magnetization rotates from one orientation to another. In a thin film, the wall's width is not arbitrary: it results from a competition between exchange energy, which prefers neighboring magnetic moments to align, and magnetostatic energy, which prefers to reduce stray fields. The balance of these two energies defines a characteristic length scale, the exchange length, typically a few nanometers in common ferromagnets.

Two classic wall types arise from this balance. A Bloch wall rotates the magnetization in the plane of the film, while a Néel wall rotates it perpendicular to that plane. In very thin films, Néel walls are favored because they produce smaller stray fields. The crossover thickness between the two regimes is on the order of the exchange length, roughly 5 nanometers in typical materials.

In Recognition Science, the framework models this crossover using its cost function. The module defines a domain cost, a measure of mismatch, as J(m/e), where J is the framework's forced cost function, m is a magnetic scale, and e is an energy scale. The framework proves three general facts about this cost: it is zero when the two scales are equal, it is never negative for positive inputs, and the golden-ratio-derived threshold phi minus 3/2 is positive. These are formal theorems in the machine-checked library of formal theorems.

What the module does not do is prove anything specific to domain walls. The definition of domainCost uses the generic ratio m/e without connecting m and e to exchange or magnetostatic energies. The research note attached to the module records the intended direction: a Néel transition at thickness t = J(phi) * L_ex = 0.118 * 5 nm = 0.59 nm, but that number is a structural sketch, not a derived result. The three proved facts are universal properties of J applied to any positive ratio.

The practical consequence is a clear boundary between what is established and what is hoped. The framework's cost function has the right shape for a threshold: zero at balance, positive away from it, with a positive critical value. Turning that shape into a prediction about real films requires defining m and e in magnetic terms, which remains open.

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

What this page does not claim

The module does not prove any result specific to magnetic domain walls. The 0.59 nm Néel transition value is a research note, not a theorem. The framework does not derive the exchange length from first principles.

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/Materials/Domain_Wall_Width2.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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