BCC atomic packing factor derivation:
  Touch condition along body diagonal [111]:
    4r = a√3    →    r = a√3/4    →    a = 4r/√3

  APF = (number of atoms × volume per atom) / volume of unit cell
      = (2 × (4/3)π r³) / a³

  Substituting a = 4r/√3:
    a³ = (4r/√3)³ = 64r³ / (3√3)

  APF = (2 × (4/3)π r³) / (64r³ / (3√3))
      = (8π r³ / 3) × (3√3 / (64r³))
      = 8π√3 / (3×64)
      = π√3 / 8
      ≈ 0.6802  (68.02%)

Nearest-neighbour distance (contact distance = 2r):
  d_nn = 2r = a√3/2  ≈ 0.2482 nm  (for α-Fe, a = 0.2866 nm)

Second nearest-neighbour distance (face diagonal / cube edge):
  d_2nn = a ≈ 0.2866 nm  (for α-Fe)
  Note: d_2nn / d_nn = 2/√3 ≈ 1.155  — BCC has only 15% gap between 1st and 2nd neighbours
  → explains why 8+6 coordination model is sometimes used for BCC

Peierls-Nabarro (P-N) stress:
  τ_PN ≈ (2G / (1−ν)) · exp(−2πw/b)

  G  = shear modulus (Pa)
  ν  = Poisson's ratio
  b  = Burgers vector magnitude (nm)
  w  = dislocation core width (nm) ≈ a / (1−ν) for FCC, narrower for BCC

Key insight:
  BCC dislocation cores are narrow (non-planar, spreading into {110}, {112}, {123} planes)
    → w is small → exp(−2πw/b) is closer to 1 → τ_PN is large
    → τ_PN is also strongly temperature-dependent (thermal activation needed to clear barrier)

  FCC dislocation cores are wide (planar, confined to {111} planes)
    → w is large → exp(−2πw/b) → 0 → τ_PN is very small (~10⁻⁴ G)
    → τ_PN barely temperature-dependent

Temperature dependence of flow stress:
  BCC metals: σ_y increases sharply as T falls (strong thermal component)
  FCC metals: σ_y increases only modestly as T falls (athermal; grain boundary, solute only)
  → This is the physical origin of the DBTT in BCC metals.

FCC atomic packing factor derivation:
  Touch condition along face diagonal [110]:
    4r = a√2    →    r = a/(2√2)    →    a = 2√2·r

  APF = (4 × (4/3)π r³) / a³

  Substituting a = 2√2·r:
    a³ = (2√2·r)³ = 16√2·r³

  APF = (4 × (4/3)π r³) / (16√2·r³)
      = (16π r³ / 3) / (16√2·r³)
      = π / (3√2)
      ≈ 0.7405  (74.05%)  ← theoretical maximum for identical hard spheres

Coordination number proof:
  An FCC atom at a face centre has:
    · 4 neighbours in the same {111} plane (forming a square in projection)
    · 4 neighbours in the {111} plane above (forming a triangle + centre in projection)
    · 4 neighbours in the {111} plane below
    Total: 12 nearest neighbours, all at distance a/√2

Stacking sequence ABCABC:
  Layer A: atoms at positions (0,0), (1,0), (0,1), (1,1) in unit cells
  Layer B: atoms sitting in one set of triangular hollows of layer A
  Layer C: atoms sitting in the OTHER set of triangular hollows of layer A
  → 3-layer repeat before returning to original A-layer position
  → 4 distinct close-packed {111} plane orientations in FCC:
    (111), (1̄11), (11̄1), (111̄) — each with 3 slip directions → 12 slip systems

FCC slip system enumeration:
  {111} planes: (111), (1̄11), (11̄1), (111̄)  — 4 planes
  ⟨110⟩ directions per plane: 3 each            — e.g. [101̄], [01̄1], [11̄0] for (111)
  Total slip systems: 4 × 3 = 12

  Independent slip systems from the 12 (by linear algebra test): 5
  → Von Mises criterion satisfied: 5 ≥ 5  ✓ → homogeneous polycrystal deformation possible

  Critical resolved shear stress (CRSS) for {111}⟨110⟩ in FCC:
    Pure Al at RT:   CRSS ≈ 1.0 MPa    (very low; high stacking fault energy)
    Pure Cu at RT:   CRSS ≈ 0.65 MPa   (low)
    Ni-base superal: CRSS ≈ 200+ MPa   (precipitation-hardened γ' obstacles)

  Schmid factor (m):
    τ_resolved = σ · cos(φ) · cos(λ) = σ · m
    Maximum m = 0.5 (when φ = λ = 45°)
    In single crystals this gives the "easy glide" orientation

HCP ideal c/a ratio derivation:
  In an ideal HCP structure, all nearest-neighbour distances are equal (= a).
  The mid-layer atom sits above the centroid of three basal-layer atoms.
  Height of mid-layer atom above basal plane = c/2.
  Distance from mid-layer atom to corner atom = a (nearest-neighbour condition).

  Horizontal distance from mid-layer centroid to corner:
    d_horiz = a / √3    (from hexagonal geometry of 3-atom triangle)

  Applying 3D distance condition:
    a² = (a/√3)² + (c/2)²
    a² = a²/3 + c²/4
    c²/4 = a² - a²/3 = 2a²/3
    c/a = √(8/3) = 1.6330  (ideal value)

Actual c/a ratios of engineering HCP metals:
  Metal      c/a      Deviation     Dominant slip system
  Be         1.568    -3.98%        Basal + prismatic (both significant)
  Ti (α)     1.587    -2.82%        Prismatic {101̄0}⟨112̄0⟩ dominates (non-basal!)
  Zr (α)     1.593    -2.45%        Prismatic
  Mg         1.624    -0.55%        Basal ≈ ideal; some prismatic
  Co (ε)     1.623    -0.61%        Basal
  Zn         1.856    +13.7%        Basal strongly preferred; limited ductility

  Key rule: c/a below ideal (< 1.633) → prismatic slip more accessible → better ductility
            c/a above ideal (> 1.633) → basal slip more strongly favoured → poorer ductility

HCP slip system analysis:
  Basal slip: {0001}⟨112̄0⟩
    1 basal plane × 3 slip directions = 3 systems
    Independent systems: 2   (all basal systems share c-axis; cannot produce strain along c)
    → Grains with c-axis parallel to tensile axis cannot deform by basal slip → CRACK

  Prismatic slip: {101̄0}⟨112̄0⟩
    3 prismatic planes × 1 slip direction = 3 systems
    Independent systems: 2   (same Burgers vector as basal; no new c-component)
    Combined basal + prismatic: 4 independent systems (still short of 5!)

  Pyramidal slip: {101̄1}⟨112̄0⟩  (type I) or {112̄2}⟨112̄3⟩ (type II)
    Type II pyramidal provides the essential c+a type Burgers vector
    → adds a 5th independent system → Von Mises criterion met
    BUT: pyramidal CRSS >> basal CRSS (2× to 10× higher stress required)
    → pyramidal slip activates only at elevated temperature or high stress

  Practical consequence:
    RT deformation of Mg (low c+a activity): 3 independent systems → cracking at grain boundaries
    RT deformation of Ti (strong prismatic): ~4–5 independent systems → much better formability
    Ti at elevated T (pyramidal active): 5+ systems → excellent superplastic formability

  Taylor factor (M) reflects independent system availability:
    FCC (12 systems, 5 independent): M ≈ 3.06 → ductile → used directly in σ_y = M·τ_CRSS
    HCP basal only (2 independent):  M → ∞ for some grain orientations → brittle

Iron allotropy and heat treatment mechanism:
  Step 1 — Austenitise: heat above A3 (~912°C for pure Fe; lower with C additions)
    α-Fe (BCC, 0.022%C max) → γ-Fe (FCC, 2.14%C max)
    → All carbides dissolve into solid solution (carbon accommodated in large FCC oct. voids)
    → Austenite is homogeneous, single-phase, paramagnetic, workable at temperature

  Step 2a — Slow cool (furnace or air):
    γ-Fe → α-Fe + Fe₃C    (eutectoid reaction at 727°C → PEARLITE)
    Carbon diffuses to grain boundaries and forms cementite lamellae
    Result: soft, tough pearlitic/ferritic microstructure (200–280 HV for 0.4%C normalised)

  Step 2b — Rapid quench (water/oil):
    γ-Fe cannot transform diffusively → temperature falls below Ms (martensite start)
    BCC-type shear transformation without diffusion: FCC → BCT (body-centred tetragonal)
    Carbon trapped in BCT lattice: c/a increases with carbon content:
      c/a = 1 + 0.046 × (%C)     (empirical)
      0.4%C martensite: c/a ≈ 1.018; lattice severely strained → HIGH HARDNESS (~600 HV)
      0.8%C martensite: c/a ≈ 1.037; even more strained → very high hardness (~800 HV)

  Step 3 — Temper (150–650°C below A1):
    Martensite is metastable → tempered to improve toughness
    150–200°C: precipitation of ε-carbide (Fe₂.₄C); some C left in BCT
    250–350°C: retained austenite decomposes; ε-carbide → cementite
    400–650°C: cementite spheroidises; ferrite recovery; hardness drops 600→350 HV
    Result: tempered martensite — best combination of strength and toughness

Interstitial void geometry:
  TETRAHEDRAL void: surrounded by 4 atoms at vertices of a regular tetrahedron
    BCC tetrahedral void (largest in BCC):
      r_tet = (√5/2 − 1) × r_atom    [where r_atom = radius of host atom]
      For α-Fe: r_atom = 0.1241 nm
      r_tet = (√5/2 − 1) × 0.1241 = 0.1180 × 0.1241 = 0.0146 nm
      Hmm — but commonly quoted as 0.036 nm using different convention:
      r_tet (hard sphere fit) = a(√5/4 − 1/2) = 0.2866(0.559−0.5) = 0.036 nm ← engineering value

    FCC tetrahedral void:
      r_tet_FCC = (√3/2 − 1) × r_atom × √2  ... = 0.414(a/2√2) × (√3/√2 − 1)
      = 0.225 × r_atom_FCC  ← smaller than FCC octahedral!

  OCTAHEDRAL void: surrounded by 6 atoms at vertices of a regular octahedron
    FCC octahedral void (largest in FCC — carbon sits here):
      r_oct_FCC = (√2 − 1) × r_atom = 0.4142 × r_atom
      For γ-Fe: r_atom ≈ 0.1270 nm → r_oct = 0.0526 nm ≈ 0.053 nm ← engineering value

    BCC octahedral void (present but very distorted — not square, rectangular):
      r_oct_BCC = (1 − √2/2) × r_atom ≈ 0.067a ≈ 0.019 nm  ← SMALLEST void in BCC!

Summary table (hard sphere radii, nm):
  Structure    Void type       r_void (nm)    Carbon misfit ratio (r_C/r_void)
  BCC α-Fe     Tetrahedral     0.036          0.077/0.036 = 2.14  (severe)
  BCC α-Fe     Octahedral      0.019          0.077/0.019 = 4.05  (extreme)
  FCC γ-Fe     Octahedral      0.053          0.077/0.053 = 1.45  (moderate)
  FCC γ-Fe     Tetrahedral     0.029          0.077/0.029 = 2.66  (more severe than oct.)
  HCP           Octahedral      ~0.053–0.063   similar to FCC

  Carbon prefers: FCC octahedral (least misfit) >> BCC tetrahedral >> BCC octahedral
  This hierarchy directly predicts the observed solubility order: FCC >> BCC