Materials Piezo Coefficient Rs

Piezoelectric coefficients

A piezoelectric material generates an electric voltage when squeezed, and the reverse: it changes shape when a voltage is applied. The standard measure of this effect is the coefficient d33, expressed in picocoulombs per newton (pC/N). It tells you how much charge a material produces per unit of applied force. For the common ceramic lead zirconate titanate (PZT), d33 is around 600 pC/N. For barium titanate, another widely used piezoelectric ceramic, the value is nearer 85 pC/N. These numbers are measured in laboratories, and they vary with the material's composition and manufacturing history.

The coefficient is not a single universal constant. It depends on the crystal structure, the direction of the applied force relative to the crystal axes, and the temperature. A higher d33 means a more sensitive sensor or a more powerful actuator, which is why materials scientists work to increase it. The values above are typical reference points, not exact specifications for every sample.

In Recognition Science, the framework models this coefficient with a simple scaling idea. The framework defines a recognition cost, a discrete record of how expensive it is to recognize one state from another, and applies it to the ratio of two quantities it calls m and e. The framework's library, a machine-checked collection of formal theorems, proves three general facts about this cost function: it is zero when the two quantities are equal, it is never negative for positive inputs, and a certain threshold value, phi minus 1.5, is greater than zero. Here phi is the golden ratio, about 1.618, so the threshold is about 0.118.

The framework then proposes that the piezoelectric coefficient follows a phi-power ladder, where d33 equals phi raised to some integer power times 1 pC/N. For example, phi to the 13th power is about 521 pC/N, close to the PZT value of 600. Phi to the 15th power is about 1364 pC/N. These are research notes, not results. The library proves nothing specific to piezoelectricity, because the module defines its cost function without connecting m and e to any physical property of the material.

What the module actually establishes, in plain language, is a template. It shows that if you define a cost as a function of a ratio, then certain basic properties follow automatically: zero at equality, nonnegativity, and a positive threshold. What it does not establish is that this cost has anything to do with real piezoelectric materials. The connection between the abstract ratio and the measured d33 values remains an open question, a target for future work rather than a derived theorem.

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

The framework derives the measured d33 values for PZT or barium titanate. The module proves that the phi-power ladder is a correct model for piezoelectricity. The recognition cost is a physical energy or charge in the conventional sense.

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

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Derived articles

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