### ✅ **def vacancy_concentration():** Nv / N = exp(-Qv / (k * T)) - **Nv** = number of vacancies - **N** = total atomic sites - **Qv** = activation energy for vacancy formation - **k** = Boltzmann constant (8.62×10−5 eV/atom·K) - **T** = absolute temperature (Kelvin) - **→** Higher T = more vacancies --- ### ✅ **def mass_density():** ρ = (n * A) / (V * NA) - Used for finding number of atomic sites in a given volume - A = atomic weight, NA = Avogadro’s number --- ### ✅ **def substitutional_solution():** Conditions (Hume-Rothery Rules): 1. Δr < 15% (atomic radius difference) 2. Same crystal structure 3. Similar electronegativity 4. Similar valency → Predicts whether a substitutional solid solution will form --- ### ✅ **def weight_percent():** C1 = 100 * m1 / (m1 + m2) ### ✅ **def atom_percent():** C′1 = 100 * n1 / (n1 + n2) - m = mass - n = number of moles --- ### ✅ **def grain_size_linear_intercept():** l = LT / (P * M) - **l** = mean intercept length (grain size) - **LT** = total length of all lines - **P** = number of grain boundary intersections - **M** = magnification --- ### ✅ **def ASTM_grain_number():** N = 2n−1 - **N** = number of grains per in2 at 100× magnification - **n** = ASTM grain size number --- ### ✅ **def dislocation_types():** - **Edge dislocation**: extra half-plane of atoms (b ⊥ dislocation line) - **Screw dislocation**: spiral shift in atomic planes (b ‖ dislocation line) - **Mixed dislocation**: combination of edge + screw → b = Burgers vector (magnitude and direction of lattice distortion) --- ### ✅ **def interfacial_defects():** - Grain boundaries = zones between crystals - Properties: high mobility, diffusivity, reactivity - Lower atomic density → more active chemically --- ### ✅ **def stacking_fault():** - Disruption in normal atomic stacking (e.g., ABCABC → ABCABABC) - Common in FCC metals --- ### ✅ **def twin_boundary():** - Mirror reflection of atomic arrangement across a plane - Often caused by mechanical deformation --- ### ✅ **def catalysts_and_defects():** - Catalytic activity often originates from surface defects - Defects act as **active sites** in chemical reactions --- ### ✅ **def microscopy_limits():** - **Optical microscope**: ~0.1 μm (10−7 m) resolution - **Electron microscope**: atomic resolution (~0.003 nm) - **STM**: visualizes atoms directly on surfaces --- ## 🧾 Summary of Defects | Type | Description | Examples | |-----------------|------------------------------------------|---------------------------------| | **Point** | 0D: vacancies, interstitials, substitutionals | atoms missing or misplaced | | **Line** | 1D: dislocations (edge, screw) | cause plastic deformation | | **Area** | 2D: grain boundaries, stacking faults | influence diffusion & strength | | **Volume** | 3D: voids, inclusions | affect mechanical integrity |