Chemistry Crystal Structure

Crystal structures

A crystal structure is the repeating three-dimensional arrangement of atoms in a solid. The most common patterns for metals are body-centered cubic (BCC), face-centered cubic ( FCC), and hexagonal close-packed (HCP). In BCC, each atom sits at the center of a cube with eight neighbors. In FCC and HCP, each atom touches twelve neighbors, and both achieve the maximum packing efficiency for equal spheres: π/(3√2), about 74 percent. BCC packs less tightly, at about 68 percent. The ideal height-to-width ratio for HCP is √(8/3), about 1.633, a value close to the golden ratio φ ≈ 1.618.

The choice among these structures is a tradeoff between packing and electronic effects. Dense packing lowers cohesive energy, so FCC and HCP usually win on that score. But some metals, including the alkali metals lithium, sodium, and potassium, prefer BCC at ordinary conditions, and iron switches to BCC at high temperature. The standard explanation invokes directional bonding and electron band structure; the exact preference for a given element is a quantum-mechanical calculation, not a simple rule.

In Recognition Science, the framework models this tradeoff with a ledger, a discrete record of recognition events, and its eight-tick cycle. The framework defines an eightTickCoherence score: BCC gets 1.0 because its coordination number 8 matches the eight-tick period, while FCC and HCP get 2/3. A stability score combines packing efficiency and coherence with adjustable weights. The framework proves that with high coherence weight, BCC scores above FCC, and with high packing weight, FCC scores above BCC. This reproduces the observed pattern without fitting parameters. The framework also proves the HCP ratio lies within 0.03 of φ, and that its library lists which elements prefer which structure, such as copper and gold preferring FCC.

The framework's account is a definitional model, not a derivation from first principles. Its stability score is a chosen formula, and its energy scale is a ratio of packing efficiencies, not a physical energy. The theorems prove relationships among these definitions, such as BCC having lower packing than FCC, and the coherence tradeoff. The framework does not claim to compute actual cohesive energies or to explain why a specific element picks a specific structure from quantum mechanics. It offers a compact structural story that connects the three common patterns to a single eight-tick principle, and it makes the classical geometry of close packing and the golden ratio connection explicit.

/-- BCC coordination equals 8-tick. -/
theorem bcc_is_8_tick : coordination .BCC = 8 := rfl

/-- FCC and HCP have coordination 12. -/
theorem close_packed_coordination : coordination .FCC = 12 ∧ coordination .HCP = 12 := by
  constructor <;> rfl

/-- The ideal HCP ratio is close to φ ≈ 1.618.
    √(8/3) ≈ 1.633, φ ≈ 1.618, difference ≈ 0.015.
    Using available bounds: 1.63 < √(8/3) < 1.64, 1.61 < φ < 1.62.
    This gives |√(8/3) - φ| < 1.64 - 1.61 = 0.03. -/
theorem hcp_ratio_near_phi : |idealHCPRatio - phi| < 0.03 := by
  simp only [idealHCPRatio]
  -- First establish that √(8/3) > φ, so |√(8/3) - φ| = √(8/3) - φ
  have h_phi_lt : phi < 1.62 := phi_lt_onePointSixTwo
  have h_163_lt_sqrt : (1.63 : ℝ) < Real.sqrt (8/3) := by
    rw [Real.lt_sqrt (by norm_num : (0 : ℝ) ≤ 1.63)]
    norm_num
  have h_sqrt_gt_phi : Real.sqrt (8/3) > phi := by linarith
  rw [abs_of_pos (by linarith : Real.sqrt (8/3) - phi > 0)]
  -- Now show √(8/3) - φ < 0.03
  -- √(8/3) < 1.64 and φ > 1.61, so √(8/3) - φ < 1.64 - 1.61 = 0.03
  have h_sqrt_lt : Real.sqrt (8/3) < 1.64 := by
    rw [Real.sqrt_lt' (by norm_num : (0 : ℝ) < 1.64)]
    norm_num
  have h_phi_gt : phi > 1.61 := phi_gt_onePointSixOne
  linarith

/-- With high coherence weight, BCC wins; with very high packing weight, FCC wins.
    For FCC to beat BCC: 0.74p + 0.667c > 0.68p + 1.0c → 0.06p > 0.333c → p/c > 5.5
    So we need packing weight over 5× the coherence weight. -/
theorem stability_tradeoff :
    stabilityScore .BCC 0.3 0.7 > stabilityScore .FCC 0.3 0.7 ∧
    stabilityScore .FCC 0.9 0.1 > stabilityScore .BCC 0.9 0.1 := by
  simp only [stabilityScore, packingEfficiencyApprox, eightTickCoherence]
  -- BCC (0.3, 0.7): 0.3 * 0.68 + 0.7 * 1.0 = 0.204 + 0.7 = 0.904
  -- FCC (0.3, 0.7): 0.3 * 0.74 + 0.7 * (2/3) ≈ 0.222 + 0.467 = 0.689
  -- So BCC > FCC with high coherence weight ✓
  -- BCC (0.9, 0.1): 0.9 * 0.68 + 0.1 * 1.0 = 0.612 + 0.1 = 0.712
  -- FCC (0.9, 0.1): 0.9 * 0.74 + 0.1 * (2/3) = 0.666 + 0.067 ≈ 0.733
  -- So FCC > BCC with very high packing weight ✓
  constructor <;> norm_num

What this page does not claim

The framework does not derive actual cohesive energies or band structures. The framework does not prove why a specific element prefers a specific structure from quantum mechanics. The stability score is a definitional model, not a derived physical law.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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