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Lean: update Semantics modules, add new numerics/physics data files Hardware: update FPGA bitstreams (tangnano9k_uart_loopback) Infra: k3s-flake tests, netcup-vps configuration, VCN compute substrate Docs: ARCHITECTURE, specs, citation updates
211 lines
10 KiB
Text
211 lines
10 KiB
Text
/-
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DrexlerianMechanosynthesis.lean — Atomic Building: STM, Tunneling, Morse Potential
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This module formalizes the mathematical models governing atomically precise
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manufacturing, from Drexler's 1986 theory to the 2026 experimental realization.
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The three core models:
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1. Tersoff-Hamann tunneling current (positioning)
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2. Morse potential (bond energy landscape)
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3. Bell-Evans-Polanyi principle (force-modified reaction rates)
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Key insight: These same structures appear at every scale:
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• Atomic: STM mechanosynthesis
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• Molecular: polymer mechanochemistry
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• Nuclear: alpha decay (Gamow factor)
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• Cosmic: false vacuum decay (instantons)
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References:
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- Drexler, K.E. (1986) "Engines of Creation"
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- arXiv:2605.27250 — Atomically precise mechanosynthesis (2026)
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- Tersoff & Hamann (1985) — STM tunneling theory
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- Morse, P.M. (1929) — Diatomic potential
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- Bell, G.I. (1978) — Models for elastically forced bonds
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Part of the OTOM TreeDIAT/PIST family.
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-/
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import Semantics.FixedPoint
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import Semantics.PIST.Spectral
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import Semantics.Q16_16Numerics
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namespace Semantics.DrexlerianMechanosynthesis
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open Semantics.Q16_16
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open Semantics.Q16_16Numerics
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open Semantics.PIST.Spectral
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §1 TERSOFF-HAMANN TUNNELING CURRENT (positioning model)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- The tunneling current between STM tip and sample.
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I ∝ V · ρ_s(E_F) · e^(-2κz)
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Where:
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V = bias voltage
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ρ_s(E_F) = local density of states at Fermi level
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z = tip-sample distance
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κ = decay constant = √(2mφ)/ℏ -/
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structure TunnelingCurrent where
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V : Q16_16 -- bias voltage (V)
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rho_s : Q16_16 -- local density of states (states/eV)
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z : Q16_16 -- tip-sample distance (Å)
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kappa : Q16_16 -- decay constant (Å⁻¹)
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deriving Repr
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/-- Compute tunneling current from parameters.
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I = V · ρ_s · exp(-2κz)
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Uses rigorous Q16_16Numerics.exp for the exponential. -/
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def computeTunnelingCurrent (tc : TunnelingCurrent) : Q16_16 :=
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-- I = V · ρ_s · exp(-2κz)
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let exponent := Q16_16.neg (Q16_16.mul (Q16_16.ofRawInt 131072) (Q16_16.mul tc.kappa tc.z))
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let tunneling_factor := exp exponent
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Q16_16.mul (Q16_16.mul tc.V tc.rho_s) tunneling_factor
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/-- The decay constant κ = √(2mφ)/ℏ.
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For typical work functions (φ ≈ 4-5 eV): κ ≈ 1 Å⁻¹ -/
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def computeDecayConstant (work_function : Q16_16) : Q16_16 :=
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-- κ = √(2mφ)/ℏ ≈ 0.512 √(φ [eV]) Å⁻¹
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let sqrt_phi := Q16_16.sqrt work_function
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Q16_16.mul (Q16_16.ofRawInt 33554) sqrt_phi -- 0.512 in Q16_16
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §2 MORSE POTENTIAL (bond energy landscape)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- The Morse potential for diatomic interaction.
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V(r) = D_e [(1 - e^(-a(r-r_e)))² - 1]
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Where:
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D_e = well depth (bond dissociation energy)
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r_e = equilibrium bond distance
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a = width parameter (related to force constant) -/
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structure MorsePotential where
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D_e : Q16_16 -- dissociation energy (eV)
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r_e : Q16_16 -- equilibrium distance (Å)
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a : Q16_16 -- width parameter (Å⁻¹)
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deriving Repr
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/-- Evaluate Morse potential at distance r.
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V(r) = D_e [(1 - exp(-a(r-r_e)))² - 1]
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Uses rigorous Q16_16Numerics.exp for the exponential. -/
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def morseEvaluate (mp : MorsePotential) (r : Q16_16) : Q16_16 :=
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-- V(r) = D_e [(1 - exp(-a(r-r_e)))² - 1]
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let dr := Q16_16.sub r mp.r_e
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let exp_term := exp (Q16_16.neg (Q16_16.mul mp.a dr))
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let one_minus := Q16_16.sub Q16_16.one exp_term
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let squared := Q16_16.mul one_minus one_minus
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Q16_16.mul mp.D_e (Q16_16.sub squared Q16_16.one)
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/-- The mechanical force from Morse potential: F = -dV/dr.
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F = -2 D_e a (1 - exp(-a(r-r_e))) exp(-a(r-r_e))
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Uses rigorous Q16_16Numerics.exp for the exponential. -/
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def morseForce (mp : MorsePotential) (r : Q16_16) : Q16_16 :=
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-- F = -2 D_e a (1 - exp(-a(r-r_e))) exp(-a(r-r_e))
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let dr := Q16_16.sub r mp.r_e
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let exp_term := exp (Q16_16.neg (Q16_16.mul mp.a dr))
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let one_minus := Q16_16.sub Q16_16.one exp_term
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let neg_two_da := Q16_16.mul (Q16_16.ofRawInt (-131072)) (Q16_16.mul mp.D_e mp.a)
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Q16_16.mul (Q16_16.mul neg_two_da one_minus) exp_term
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §3 BELL-EVANS-POLANYI PRINCIPLE (force-modified reaction rates)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- The force-modified Arrhenius equation.
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k(F) = A · exp(-(E_a - F·Δx‡) / k_B T)
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Where:
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A = pre-exponential factor
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E_a = activation energy barrier
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F = applied mechanical force
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Δx‡ = activation length (distance to transition state)
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k_B T = thermal energy -/
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structure BellEvansPolanyi where
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A : Q16_16 -- pre-exponential factor (s⁻¹)
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E_a : Q16_16 -- activation energy (eV)
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delta_x : Q16_16 -- activation length (Å)
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kT : Q16_16 -- thermal energy k_B·T (eV)
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deriving Repr
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/-- Compute reaction rate under applied force.
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Simplified: k(F) ≈ A · (1 - (E_a - F·Δx‡)/k_B T) for small barriers -/
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def computeReactionRate (bep : BellEvansPolanyi) (F : Q16_16) : Q16_16 :=
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-- k(F) ≈ A · (1 - (E_a - F·Δx‡)/k_B T)
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let effective_barrier := Q16_16.sub bep.E_a (Q16_16.mul F bep.delta_x)
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let reduction := Q16_16.div effective_barrier bep.kT
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Q16_16.mul bep.A (Q16_16.sub Q16_16.one reduction)
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/-- The critical force where barrier vanishes: F_crit = E_a / Δx‡. -/
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def criticalForce (bep : BellEvansPolanyi) : Q16_16 :=
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Q16_16.div bep.E_a bep.delta_x
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §4 CROSS-SCALE INVARIANCE (same math at every scale)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Physical scale identifiers. -/
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inductive PhysicalScale where
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| atomic -- STM mechanosynthesis (Å scale)
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| molecular -- polymer mechanics (nm scale)
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| nuclear -- alpha decay, quark confinement (fm scale)
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| cosmic -- false vacuum decay, QGP (Mpc scale)
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deriving Repr, DecidableEq
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/-- The universal barrier-crossing structure.
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Same math, different physical meaning at each scale. -/
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structure UniversalBarrierCrossing where
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scale : PhysicalScale
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barrier_height : Q16_16 -- E_a or equivalent (in scale-appropriate units)
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tunneling_rate : Q16_16 -- exponential decay rate
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force_coupling : Q16_16 -- how force modifies barrier
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deriving Repr
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/-- Map atomic-scale parameters to nuclear scale (alpha decay).
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The Gamow factor is the nuclear analog of STM tunneling. -/
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def atomicToNuclear (atomic : UniversalBarrierCrossing) : UniversalBarrierCrossing :=
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{ scale := .nuclear
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, barrier_height := Q16_16.mul atomic.barrier_height (Q16_16.ofRawInt 655360) -- ~10 MeV scale
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, tunneling_rate := Q16_16.div atomic.tunneling_rate (Q16_16.ofRawInt 65536) -- narrower barrier
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, force_coupling := Q16_16.mul atomic.force_coupling (Q16_16.ofRawInt 655360) }
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §5 CARBON DIMER ASSEMBLY (specific to arXiv:2605.27250)
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- The C₂ dimer parameters for Si(100) mechanosynthesis. -/
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def c2_dimer : MorsePotential :=
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{ D_e := Q16_16.ofRawInt 409600 -- ~6.3 eV (C-C bond)
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, r_e := Q16_16.ofRawInt 78643 -- ~1.20 Å (C≡C triple bond)
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, a := Q16_16.ofRawInt 196608 } -- ~3.0 Å⁻¹ (stiff bond)
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/-- The Si-C bond parameters. -/
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def si_c_bond : MorsePotential :=
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{ D_e := Q16_16.ofRawInt 327680 -- ~5.0 eV (Si-C bond)
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, r_e := Q16_16.ofRawInt 104858 -- ~1.60 Å
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, a := Q16_16.ofRawInt 131072 } -- ~2.0 Å⁻¹
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §6 EXECUTABLE WITNESSES
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-- ═══════════════════════════════════════════════════════════════════════════
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-- Tunneling current at z = 5 Å, φ = 4.5 eV
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def testTunnel : TunnelingCurrent :=
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{ V := Q16_16.one, rho_s := Q16_16.ofRawInt 32768
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, z := Q16_16.ofRawInt 327680, kappa := Q16_16.ofRawInt 65536 }
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#eval computeTunnelingCurrent testTunnel -- expect: ~exp(-10) ≈ very small
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-- Morse potential at equilibrium
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#eval morseEvaluate c2_dimer c2_dimer.r_e -- expect: 0 (at equilibrium)
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-- Morse force at equilibrium
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#eval morseForce c2_dimer c2_dimer.r_e -- expect: 0 (no force at equilibrium)
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-- Critical force for C-C bond breaking
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def testBEP : BellEvansPolanyi :=
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{ A := Q16_16.ofRawInt 655360, E_a := Q16_16.ofRawInt 409600
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, delta_x := Q16_16.ofRawInt 6553, kT := Q16_16.ofRawInt 2556 }
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#eval criticalForce testBEP -- expect: ~6.3 eV / 0.1 Å = 63 nN
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end Semantics.DrexlerianMechanosynthesis
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