Chemistry Metallic Bond Fcc Hcp Denser Than Bcc

Packing efficiency

In a metallic solid, the atoms arrange themselves in repeating patterns called crystal lattices. The three most common patterns for metals are body-centered cubic (BCC), face-centered cubic (FCC), and hexagonal close-packed (HCP). A lattice's packing efficiency is the fraction of space inside the crystal that the atoms actually fill, treating each atom as a hard sphere. BCC, where each atom sits at the center of a cube with one atom at each corner, fills about 68 percent of the space. FCC and HCP, where each atom touches twelve neighbors in the tightest possible arrangement, each fill about 74 percent. The machine-checked theorem fcc_hcp_denser_than_bcc proves the numerical fact that follows directly from these definitions: the packing efficiency of BCC is strictly less than that of both FCC and HCP.

This is a statement about geometry, not about any particular metal. It holds for any crystal that uses these ideal lattice types, regardless of the element or the size of its atoms. The theorem is proved in the framework's machine-checked library of formal theorems by evaluating the three defined efficiency values and comparing them. It does not say how any real metal actually packs, because real metals often form mixed structures, contain defects, or adopt different lattices under pressure or temperature. It also does not claim that FCC and HCP are equally dense in every sense; they have the same packing efficiency, but their unit cell shapes differ, and the theorem does not compare those shapes.

In Recognition Science, this geometric fact connects to a broader account of metallic bonding. The framework models the metallic state as a coherent system of delocalized electrons, and it notes that BCC's coordination number of 8 matches its eight-tick recognition cycle, while the close-packed structures have coordination 12. The density comparison itself, however, is a pure geometric theorem. It stands on the definitions of the three lattice types and their packing efficiencies, not on the framework's recognition-cost mechanism.

/-- FCC/HCP have higher packing efficiency than BCC. -/
theorem fcc_hcp_denser_than_bcc :
    packingEfficiency .BCC < packingEfficiency .FCC ∧
    packingEfficiency .BCC < packingEfficiency .HCP := by
  constructor <;> { simp only [packingEfficiency]; norm_num }

/-- Packing efficiency for each lattice type. -/
def packingEfficiency : LatticeType → ℝ
| .BCC => 0.68
| .FCC => 0.74
| .HCP => 0.74

What this page does not claim

The theorem does not state that any real metal adopts one of these lattice types. It does not claim FCC and HCP are identical in all properties, only that their packing efficiencies are equal. It does not derive the packing efficiencies from the framework's recognition-cost mechanism.

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

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