### ✅ **def atomic_packing_factor_SC():** APF = (1 * (4/3) * π * R3) / a3 → APF = 0.52 - Simple cubic structure, 1 atom/unit cell --- ### ✅ **def atomic_packing_factor_BCC():** APF = (2 * (4/3) * π * R3) / a3 → APF = 0.68 - Body-centered cubic (BCC), 2 atoms/unit cell - a = 4R / √3 --- ### ✅ **def atomic_packing_factor_FCC():** APF = (4 * (4/3) * π * R3) / a3 → APF = 0.74 - Face-centered cubic (FCC), 4 atoms/unit cell - a = 2√2 * R --- ### ✅ **def atomic_packing_factor_HCP():** APF = 0.74 - Hexagonal close-packed (HCP), 6 atoms/unit cell - Ideal c/a ratio = 1.633 --- ### ✅ **def theoretical_density():** ρ = (n * A) / (VC * NA) - ρ = density (g/cm3) - n = atoms per unit cell - A = atomic weight (g/mol) - VC = volume of unit cell (cm3) - NA = Avogadro’s number (6.022×1023 atoms/mol) --- ### ✅ **def linear_density():** LD = number_of_atoms_centered / length_of_vector --- ### ✅ **def planar_density():** PD = number_of_atoms_in_plane / area_of_plane --- ### ✅ **def bragg_law():** d = (n * λ) / (2 * sin(θ)) - d = interplanar spacing - λ = wavelength of incident X-ray - θ = diffraction angle --- ### ✅ **def miller_indices(h, k, l):** (hkl) = reciprocals of intercepts with axes a, b, c → Algorithm: 1. Intercepts 2. Reciprocals 3. Clear fractions 4. Write as (hkl) --- ### ✅ **def crystallographic_direction():** [uvw] → Steps: 1. Tail-to-head vector 2. Divide by unit cell dimensions 3. Reduce to smallest integers 4. Enclose in square brackets --- ### ✅ **def hcp_direction():** [uvtw] → Use 4-index Miller-Bravais system u = 1⁄2(2u′ - v′) v = 1⁄2(2v′ - u′) t = -(u + v) w = w′ --- ### ✅ **def polymorphism():** When a material has multiple crystal structures - Example: Fe (BCC ↔ FCC), Carbon (graphite ↔ diamond) --- ### ✅ **def single_vs_polycrystals():** - **Single crystals** → anisotropic - **Polycrystals** → isotropic (if random orientation) --- ### ✅ **def material_density_trends():** ρ_metals > ρ_ceramics > ρ_polymers - Metals: close-packed, heavy atoms - Ceramics: moderate packing, lighter atoms - Polymers: low packing, low mass