Chemistry Crystal Structure Bcc Packing Lt Fcc

The packing inequality

In crystallography, the packing efficiency of a structure is the fraction of space filled by the atoms themselves, modeled as hard spheres. For body-centered cubic (BCC), where atoms sit at each cube corner and one at the center, the efficiency is π√3/8, about 0.68. For face-centered cubic (FCC), with atoms at the corners and at the center of each face, it is π/(3√2), about 0.74. The theorem bcc_packing_lt_fcc in the machine-checked library of formal theorems proves the inequality: 0.68 is less than 0.74. It is a statement about geometry, not about which metal will actually form which crystal.

The distinction matters because packing efficiency correlates with cohesive energy. The same library defines an energy scale where BCC is 1.0 and FCC is 0.917, and proves that the close-packed structures (FCC and HCP) have lower energy than BCC on this scale. Yet many real metals, including the alkali metals lithium, sodium, and potassium, prefer BCC at room temperature. The framework's account explains this through an additional factor: an eight-tick coherence, a measure of how well a structure matches an eight-step recognition cycle from its underlying theory. BCC has perfect coherence (1.0), while FCC and HCP score only 2/3. A stability score combines packing efficiency and coherence with weights, and the library proves that BCC wins when coherence is weighted heavily, while FCC wins when packing dominates. This is a model, not a law of nature.

The theorem itself is narrow. It does not predict that any specific element will choose BCC or FCC. It does not establish that BCC is always less stable than FCC, only that its geometric packing fraction is smaller. The library does define lists of elements that prefer each structure, and proves that the alkali metals with atomic numbers 3, 11, 19, 37, and 55 appear on the BCC list, but those are separate definitions and decisions, not consequences of the packing inequality. The inequality is a single, verified fact about two numbers, useful as a building block for the larger stability story.

/-- BCC has lower packing than FCC/HCP. -/
theorem bcc_packing_lt_fcc : packingEfficiencyApprox .BCC < packingEfficiencyApprox .FCC := by
  simp only [packingEfficiencyApprox]
  norm_num

/-- Packing efficiency (fraction of space filled by spheres). -/
def packingEfficiency : Structure → ℝ
| .BCC => Real.pi * Real.sqrt 3 / 8  -- ≈ 0.68
| .FCC => Real.pi / (3 * Real.sqrt 2) -- ≈ 0.74
| .HCP => Real.pi / (3 * Real.sqrt 2) -- ≈ 0.74

/-- Packing efficiency (fraction of space filled by spheres). -/
def packingEfficiency : Structure → ℝ
| .BCC => Real.pi * Real.sqrt 3 / 8  -- ≈ 0.68
| .FCC => Real.pi / (3 * Real.sqrt 2) -- ≈ 0.74
| .HCP => Real.pi / (3 * Real.sqrt 2) -- ≈ 0.74

/-- BCC is favored when 8-tick coherence dominates.
    The coordination number 8 directly reflects ledger periodicity. -/
def eightTickCoherence : Structure → ℝ
| .BCC => 1.0      -- Perfect 8-tick match
| .FCC => 2/3      -- 8/12 = 2/3 match
| .HCP => 2/3      -- Same as FCC

/-- 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 theorem does not predict which specific elements will form BCC or FCC structures. The packing inequality alone does not establish that BCC is less stable than FCC. The energy scale is a definitional model, not a derived physical law.

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