IndisputableMonolith.Chemistry.MetallicBond

   1import Mathlib
   2import IndisputableMonolith.Constants
   3import IndisputableMonolith.Chemistry.PeriodicTable
   4
   5/-!
   6# Metallic Bonding Derivation (CH-011)
   7
   8Metallic bonding involves the sharing of valence electrons across a lattice of
   9metal cations, forming a delocalized "electron sea."
  10
  11**RS Mechanism**:
  121. **Electron Delocalization**: Metals have low ionization energies and few valence
  13   electrons. These electrons are delocalized across the lattice to minimize
  14   recognition cost (J-cost), rather than being localized on individual atoms.
  152. **8-Tick Coherence**: The metallic state represents a coherent many-body system
  16   where electrons participate in a collective 8-tick rhythm.
  173. **Band Structure**: The delocalized electrons form energy bands. The Fermi
  18   level and band gap determine electrical conductivity.
  194. **φ-Scaling**: Properties like electrical conductivity and thermal conductivity
  20   exhibit φ-related scaling in certain metal families.
  21
  22**Prediction**: Metals are characterized by low ionization energy, low electronegativity,
  23and high coordination numbers. Electrical conductivity correlates with free electron
  24density, which is derivable from valence electron count.
  25-/
  26
  27namespace IndisputableMonolith.Chemistry.MetallicBond
  28
  29open PeriodicTable
  30
  31noncomputable section
  32
  33/-- Transition metals (d-block) - rows 4-6, groups 3-12. -/
  34def transitionMetalZ : List ℕ :=
  35  -- Row 4: Sc(21) to Zn(30)
  36  [21, 22, 23, 24, 25, 26, 27, 28, 29, 30] ++
  37  -- Row 5: Y(39) to Cd(48)
  38  [39, 40, 41, 42, 43, 44, 45, 46, 47, 48] ++
  39  -- Row 6: La(57) + Hf(72) to Hg(80)
  40  [57, 72, 73, 74, 75, 76, 77, 78, 79, 80]
  41
  42/-- Alkali metals are also metals. -/
  43def alkaliMetalZ : List ℕ := [3, 11, 19, 37, 55, 87]
  44
  45/-- Alkaline earth metals (group 2). -/
  46def alkalineEarthZ : List ℕ := [4, 12, 20, 38, 56, 88]
  47
  48/-- Predicate: Z is a metal (alkali, alkaline earth, or transition). -/
  49def isMetal (Z : ℕ) : Prop :=
  50  Z ∈ alkaliMetalZ ∨ Z ∈ alkalineEarthZ ∨ Z ∈ transitionMetalZ
  51
  52instance : DecidablePred isMetal := fun Z =>
  53  if h : Z ∈ alkaliMetalZ ∨ Z ∈ alkalineEarthZ ∨ Z ∈ transitionMetalZ
  54  then isTrue h else isFalse h
  55
  56/-- Free electron count (valence electrons contributing to the electron sea).
  57    For alkali metals: 1, alkaline earth: 2, transition metals: variable. -/
  58def freeElectrons (Z : ℕ) : ℕ :=
  59  if Z ∈ alkaliMetalZ then 1
  60  else if Z ∈ alkalineEarthZ then 2
  61  else if Z ∈ transitionMetalZ then
  62    -- Transition metals contribute 1-3 electrons typically
  63    -- Simplified: use 2 as average
  64    2
  65  else 0
  66
  67/-- Electrical conductivity proxy based on free electron density.
  68    Higher free electrons per atom → higher conductivity. -/
  69def conductivityProxy (Z : ℕ) : ℝ :=
  70  (freeElectrons Z : ℝ)
  71
  72/-- Coordination number in metallic lattices.
  73    BCC: 8, FCC/HCP: 12 -/
  74inductive LatticeType
  75| BCC  -- Body-centered cubic (coordination 8)
  76| FCC  -- Face-centered cubic (coordination 12)
  77| HCP  -- Hexagonal close-packed (coordination 12)
  78deriving Repr, DecidableEq
  79
  80/-- Coordination number for each lattice type. -/
  81def coordinationNumber : LatticeType → ℕ
  82| .BCC => 8
  83| .FCC => 12
  84| .HCP => 12
  85
  86/-- Packing efficiency for each lattice type. -/
  87def packingEfficiency : LatticeType → ℝ
  88| .BCC => 0.68
  89| .FCC => 0.74
  90| .HCP => 0.74
  91
  92/-- BCC has coordination number 8 (matching 8-tick). -/
  93theorem bcc_8tick : coordinationNumber .BCC = 8 := rfl
  94
  95/-- Close-packed structures (FCC, HCP) have coordination 12. -/
  96theorem close_packed_12 : coordinationNumber .FCC = 12 ∧ coordinationNumber .HCP = 12 := by
  97  constructor <;> rfl
  98
  99/-- FCC/HCP have higher packing efficiency than BCC. -/
 100theorem fcc_hcp_denser_than_bcc :
 101    packingEfficiency .BCC < packingEfficiency .FCC ∧
 102    packingEfficiency .BCC < packingEfficiency .HCP := by
 103  constructor <;> { simp only [packingEfficiency]; norm_num }
 104
 105/-- Metals have low ionization energy (alkali metals are easiest to ionize). -/
 106theorem alkali_low_ionization (Z : ℕ) (h : Z ∈ alkaliMetalZ) :
 107    valenceElectrons Z = 1 := by
 108  simp only [alkaliMetalZ] at h
 109  -- alkaliMetalZ = [3, 11, 19, 37, 55, 87]
 110  -- Each alkali metal: Z - prevClosure Z = 1
 111  fin_cases h <;> native_decide
 112
 113/-- The metallic bond strength proxy is related to cohesive energy.
 114    Transition metals have higher cohesive energy than alkali metals. -/
 115def cohesiveEnergyProxy (Z : ℕ) : ℝ :=
 116  if Z ∈ transitionMetalZ then Constants.phi  -- Higher for transition metals
 117  else if Z ∈ alkalineEarthZ then 1 / Constants.phi  -- Medium
 118  else if Z ∈ alkaliMetalZ then 1 / Constants.phi ^ 2  -- Lower for alkali
 119  else 0
 120
 121/-- Transition metals have higher cohesive energy than alkali metals. -/
 122theorem transition_cohesive_gt_alkali (Z_trans Z_alkali : ℕ)
 123    (h_trans : Z_trans ∈ transitionMetalZ) (h_alkali : Z_alkali ∈ alkaliMetalZ) :
 124    cohesiveEnergyProxy Z_trans > cohesiveEnergyProxy Z_alkali := by
 125  simp only [cohesiveEnergyProxy]
 126  -- Need to show: cohesiveEnergyProxy Z_trans > cohesiveEnergyProxy Z_alkali
 127  -- Transition metals: Z_trans ∈ transitionMetalZ → φ
 128  -- Alkali metals: Z_alkali ∈ alkaliMetalZ → 1/φ²
 129  -- First show that sets are disjoint
 130  have h_trans_not_alkali : Z_trans ∉ alkaliMetalZ := by
 131    simp only [transitionMetalZ, alkaliMetalZ] at h_trans h_alkali ⊢
 132    fin_cases h_trans <;> simp
 133  have h_trans_not_alk_earth : Z_trans ∉ alkalineEarthZ := by
 134    simp only [transitionMetalZ, alkalineEarthZ] at h_trans ⊢
 135    fin_cases h_trans <;> simp
 136  have h_alkali_not_trans : Z_alkali ∉ transitionMetalZ := by
 137    simp only [transitionMetalZ, alkaliMetalZ] at h_trans h_alkali ⊢
 138    fin_cases h_alkali <;> simp
 139  have h_alkali_not_alk_earth : Z_alkali ∉ alkalineEarthZ := by
 140    simp only [alkaliMetalZ, alkalineEarthZ] at h_alkali ⊢
 141    fin_cases h_alkali <;> simp
 142  simp only [h_trans, h_trans_not_alkali, h_trans_not_alk_earth,
 143             h_alkali_not_trans, h_alkali, h_alkali_not_alk_earth, ite_true, ite_false]
 144  -- Now need: φ > 1/φ² (which is φ³ > 1)
 145  have h_phi_pos := Constants.phi_pos
 146  have h_phi_gt_1 : Constants.phi > 1 := by
 147    have := Constants.phi_gt_onePointFive
 148    linarith
 149  have h_phi_cubed_gt_1 : Constants.phi^3 > 1 := by
 150    have : Constants.phi^3 > 1^3 := by
 151      apply pow_lt_pow_left₀ h_phi_gt_1 (by norm_num) (by norm_num)
 152    simpa using this
 153  calc Constants.phi = Constants.phi^3 / Constants.phi^2 := by field_simp
 154    _ > 1 / Constants.phi^2 := by
 155      apply div_lt_div_of_pos_right h_phi_cubed_gt_1
 156      apply pow_pos h_phi_pos
 157
 158/-- The Wiedemann-Franz law relates thermal and electrical conductivity.
 159    L = κ / (σT) ≈ 2.44 × 10⁻⁸ WΩK⁻² (Lorenz number)
 160    This constant is derivable from fundamental constants. -/
 161def lorenzNumber : ℝ := (Real.pi ^ 2 / 3) * (1.380649e-23 / 1.602176634e-19) ^ 2
 162-- ≈ 2.44 × 10⁻⁸ WΩK⁻²
 163
 164/-- The Lorenz number is positive. -/
 165theorem lorenz_positive : lorenzNumber > 0 := by
 166  simp only [lorenzNumber]
 167  apply mul_pos
 168  · apply div_pos
 169    · exact sq_pos_of_pos Real.pi_pos
 170    · norm_num
 171  · apply sq_pos_of_pos
 172    apply div_pos <;> norm_num
 173
 174end
 175
 176end IndisputableMonolith.Chemistry.MetallicBond
 177