Unit Cell Atoms (Z)

$Z_{SC} = 1 \qquad Z_{BCC} = 2 \qquad Z_{FCC} = 4$

Edge–Radius Relations

$SC: r = \frac{a}{2} \qquad BCC: r = \frac{a\sqrt{3}}{4} \qquad FCC: r = \frac{a\sqrt{2}}{4}$

Packing Efficiency

$SC = 52.4\% \qquad BCC = 68\% \qquad FCC = HCP = 74\%$

Coordination Numbers

$SC = 6 \qquad BCC = 8 \qquad FCC = HCP = 12$

Density Formula

$\boxed{\rho = \frac{Z \cdot M}{a^3 \cdot N_A}}$

• Z = atoms/cell; M = molar mass (g/mol); a = edge length (cm); N_A = $6.022 \times 10^{23}$ $mol^{-1}$
• Unit conversions: 1 Å = 10^{-8} cm; 1 pm = 10^{-1}^{0} cm

Voids (for n close-packed atoms)

$\text{Octahedral voids} = n \qquad \text{Tetrahedral voids} = 2n \qquad \text{Total} = 3n$

$r^+/r^- \text{ (tet)} = 0.225 \qquad r^+/r^- \text{ (oct)} = 0.414$

Ionic Structure CN

$NaCl = 6:6 \qquad CsCl = 8:8 \qquad ZnS = 4:4 \qquad CaF_2 = 8:4$

Crystal Defects

Semiconductor Doping

$\text{Group 15 (P, As)} \rightarrow n\text{-type (electrons)} \qquad \text{Group 13 (B, Ga)} \rightarrow p\text{-type (holes)}$

Magnetic Behaviour

$Fe, Co, Ni \rightarrow \text{Ferromagnetic}$ $MnO \rightarrow \text{Antiferromagnetic}$ $Fe_3O_4 \rightarrow \text{Ferrimagnetic}$ $O_2 \rightarrow \text{Paramagnetic}$ $NaCl, C_6H_6 \rightarrow \text{Diamagnetic}$

Critical NEET Facts

1. AgBr shows BOTH Schottky and Frenkel defects
2. CsCl structure ≠ BCC (different ions at corner vs body centre)
3. ZnS fills HALF tetrahedral voids; NaCl fills ALL octahedral voids
4. F-centres absorb visible light → crystals appear coloured
5. Semiconductor conductivity increases with T; metal conductivity decreases with T