Chemistry Crystal Structure Ideal Hcp Ratio Value

The ideal ratio

In a hexagonal close-packed (HCP) crystal, atoms are arranged in layers where each atom touches twelve neighbors. The structure's height relative to its base width is called the c/a ratio. For perfect spheres packed this way, geometry fixes this ratio at the square root of 8/3, which is approximately 1.633. This is a standard result of classical crystallography, independent of any particular material.

The value is often compared to the golden ratio, phi, which is about 1.618. The two numbers are close but not identical. The ideal HCP ratio is about 0.9 percent larger than phi. Real crystals deviate from the ideal value when bonding is not perfectly spherical, a fact that makes the ratio a useful measure of anisotropy in materials science.

In Recognition Science, the framework's machine-checked library of formal theorems includes a definition of the ideal HCP ratio and a theorem about it. The definition sets the ratio to the square root of 8/3. A separate theorem, named ideal_hcp_ratio_value, proves that this value lies strictly between 1.63 and 1.64. This is a precise, formal statement of the classical geometric fact, not a new physical prediction.

The framework also proves that this ideal ratio is within 0.03 of the golden ratio phi. This proximity is presented as part of a broader pattern in the framework, where phi appears in various structural contexts. However, the theorem does not claim that the HCP ratio equals phi, nor does it derive the ratio from phi. The two numbers remain distinct, and the framework's theorem merely records their closeness.

What the declaration does not do is more important than what it does. It does not claim that any real metal has this exact ratio, nor does it predict which elements will form HCP crystals. Those questions involve the energy scales and stability scores that the framework models separately, with their own definitions and theorems. The ideal ratio is a geometric baseline, a number that describes perfect packing, not a measurement of any actual substance.

/-- Ideal c/a ratio for HCP: √(8/3) ≈ 1.633. -/
def idealHCPRatio : ℝ := Real.sqrt (8/3)

/-- The ideal HCP c/a ratio is approximately 1.633. -/
theorem ideal_hcp_ratio_value : 1.63 < idealHCPRatio ∧ idealHCPRatio < 1.64 := by
  -- √(8/3) ≈ 1.6329931..., so 1.63 < √(8/3) < 1.64
  -- We verify: 1.63² = 2.6569 < 8/3 ≈ 2.6667 < 2.6896 = 1.64²
  simp only [idealHCPRatio]
  have h_sq_lo : (1.63 : ℝ)^2 < 8/3 := by norm_num
  have h_sq_hi : (8 : ℝ)/3 < (1.64 : ℝ)^2 := by norm_num
  constructor
  · -- 1.63 < √(8/3) ⟺ 1.63² < 8/3 (for positive values)
    rw [Real.lt_sqrt (by norm_num : (0 : ℝ) ≤ 1.63)]
    exact h_sq_lo
  · -- √(8/3) < 1.64 ⟺ 8/3 < 1.64² (for positive values)
    rw [Real.sqrt_lt' (by norm_num : (0 : ℝ) < 1.64)]
    exact h_sq_hi

/-- 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

What this page does not claim

The theorem does not claim that any real metal has this exact ratio. The theorem does not predict which elements will form HCP crystals. The theorem does not claim the HCP ratio equals the golden ratio.

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)

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