Materials Sputter Yield From Jcost

Sputter yield and the cost function

Sputtering is the process where energetic ions strike a solid target and knock atoms loose from its surface. It is the workhorse of thin-film manufacturing, used to coat optics, semiconductors, and hard coatings. The key quantity is the sputter yield, the average number of target atoms ejected per incoming ion. A higher yield means faster coating and less damage to the target. The yield depends strongly on the ion's energy, rising from zero at a threshold energy E_th and climbing as the energy increases.

A standard empirical observation is that near this threshold, the yield grows roughly as the square root of how far the energy exceeds the threshold: S ~ (E/E_th - 1)^0.5. The framework's cost function J, which is forced by five plain conditions to equal J(x) = (x + 1/x)/2 - 1, offers a specific form for this relationship. In Recognition Science, the module models the yield as S = J(phi) * (E/E_th - 1)^0.5 near threshold, where phi is the golden ratio, approximately 1.618. At the special energy E = phi * E_th, this gives S = J(phi)^0.5 * phi^0.25, which evaluates to roughly 0.35.

The formal content is deliberately modest. It proves three general facts about the cost function J(m/e), where m and e are positive real numbers: the cost vanishes when m equals e, the cost is never negative for positive inputs, and the quantity phi - 3/2 is positive. These facts are collected into a certificate structure. The module does not prove anything specific about sputtering, because the definition of domainCost as J(m/e) does not yet tie m and e to physical quantities like ion mass or energy. The research note in the file records where the idea was meant to go, not a result that has been established.

What this means in plain language is that the framework has a candidate formula for sputter yield, but the physical bridge is not yet built. The formal library proves the mathematics of the cost function itself, including that it is well-behaved and positive, and that a natural threshold quantity is positive. What would turn this into a theorem about sputtering is a definition of m and e in the subject's own terms, connecting them to actual ion and target properties. Until then, the sputter yield formula remains a research note, not a derived result.

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 the sputter yield formula S = J(phi) * (E/E_th - 1)^0.5 as a theorem about physical sputtering. The module does not establish that m and e correspond to any specific physical quantities such as ion mass or energy. The numerical value 0.35 for the yield at E = phi * E_th is a research note, not a measured or derived result.

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

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