Materials Nanoparticle Size From Jcost

Quantum confinement threshold

Quantum confinement is the size effect that changes a semiconductor nanoparticle's optical and electronic properties. When a particle's diameter d shrinks below about twice the Bohr radius a_B of the exciton, the electron-hole pair can no longer fit comfortably inside, so its energy levels become discrete and the material's band gap widens. The classical onset condition is d < 2 a_B, and the transition is not gradual: below that size, the particle behaves less like a bulk crystal and more like a giant artificial atom.

The Bohr radius itself is a length scale from hydrogen-like physics, typically a few nanometers in common semiconductors like cadmium selenide. For a nanoparticle, the ratio d / a_B is the natural dimensionless measure of confinement. When that ratio is small, confinement is strong; when it is large, the particle behaves like bulk material. This ratio is exactly the kind of quantity that a cost function in Recognition Science can evaluate.

In Recognition Science, the framework models physical transitions as points where a recognition cost crosses a threshold. The cost function J(x) = (x + 1/x)/2 - 1 measures the mismatch between two quantities, vanishing when they are equal and growing as they diverge. The framework's golden ratio phi, about 1.618, appears as the self-similar scaling that the cost function forces. Applying J to the ratio d / a_B and setting it equal to J(phi) gives the confinement threshold d = phi × 2 a_B, roughly 1.618 times the Bohr diameter.

The machine-checked library of formal theorems in the framework proves three general facts about this cost function: it vanishes when the two inputs are equal, it is nonnegative for positive inputs, and the constant phi - 3/2 is positive. These facts are stated universally, not specifically for nanoparticles. The module defines a certificate structure that packages these three properties, and proves that such a certificate exists. What the module does not do is define the particle diameter m and the Bohr radius e in terms of any specific semiconductor's physics; that definition remains a research note, not a theorem.

The practical consequence is a clean rule of thumb: a nanoparticle enters the quantum confinement regime when its diameter is about 1.6 times the Bohr diameter. For a material with a Bohr radius of 5 nanometers, that means particles below roughly 8 nanometers start showing size-tunable optical properties. The framework's contribution is not a new measurement but a structural reason why the golden ratio appears at the transition, and an honest statement of what remains to be connected to real materials.

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]

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

What this page does not claim

No specific semiconductor's Bohr radius is defined in the module. The quantum confinement threshold is not a proved theorem about nanoparticles, only a research note. The framework does not derive the Bohr radius 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/NanoparticleSizeFromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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